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Nittaya Noinan
Nittaya Noinan
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40 1. - 1. x + 3y x – 2y = = 8 3 …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. ……………………………………………………………………
2. x- y = 2x + 3y = -4 22 …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. 3. x+ y+z = 6 …………………………………………………………………… x–y+z = 2 ……………………………. = x+y-z 0
4. x + 2y -z = 3 3x + y = 6 2x + y = 1 …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. ……………………………………………………………………
5. 2x - 3y + z = 8 - x + 4y + 2z = -4 3x - y + 2z = 9 …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… …………………………….
6. x - y + 2z = -3 y – 3z = 5 x + 4y – 8z = 17 …………………………………………………………………… ……………………………. …………………………………………………………………… …………………………….
7. x + y = 2 x + 3y + z = 5 3x + y - z = 3 …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… …………………………….
8. 2x + 2y + 3z + 2t = -3 x + y + 2z + 2t = 6 2y + 5z + 2t = 5 x + y + 3z + 4t = 1 …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. ……………………………………………………………………
2 1 1. A = B = 3 5 9 3. C = 2 5 0 1 2. 1 1 3 1 3 5 3 4
2 2
0 1. 3 2 A9 a 4 5 4 1 a13 = a21 = A23 = a11 = 1 0 5 4 1 3 A2 2 4 7 5 1 3 a 6 9 8 3 2. a14 = a21 = A32 = a12 = A14 = a35 =
3 1. 1. 4 1 7 9 …………………….. 3 1 1 3 2. 0 2 2 1 0 0 3 0 0 0 0 6 …………………….. 3. 2 3 4 …………………….. 4. 2 0 0 2 …………………….. 5. 0 0 0 0 0 0 ……………………..
2. 1) 2 3 2 2) 0 1 4 2 5 3) 1 0 0 1 9) 6 2 0 3 0 0 1 4) 0 0 0 0 5) 0 1 1 0 6) 1 0 0 0 1 0 0 0 1 7) 0 0 0 8) 0 0 0 0 0 5 12) 0 1 8 . 10) 5 0 1 2 14) 0 0 0 15) 9 0 0 8 5 0 3 0 2 0 0 0 2 3 13) 2 0 0 1 5 11) 0 .
……………………………………………………………… …………………... ……………………………………………………………… …………………... ……………………………………………………………… ………………….. ……………………………………………………………… …………………... 4
1. A 1 2 2x 6 1 , B A = B y 8 3 2 x y ....................................................................................... ............................ ……………………………………………………………… …………… ……………………………………………………………… 3x y 2 1 2 2. …………… A x = A = B 2y , B 5 y = 10 x 5 y ....................................................................................... ............................ ……………………………………………………………… …………… ……………………………………………………………… 3. …………… x = A = B A o 4 3 8 y = , B x x x 3 y 8 y x = …………… y = …………… .......................................................................................
3 4. A 1 4 2 0 1 x 4 0 8 2 5 8 , B x 5 A = B 3 4 4 x y y ....................................................................................... ............................ ……………………………………………………………… …………… ……………………………………………………………… 1 9 2 3 3x 2 x y 5. …………… 3 4 5 2 5 x = A 4 y = A = B , B 3 4 5 2 5 4 x y ....................................................................................... ............................ ……………………………………………………………… ……………
6. A x y x y A = B 1 , B 7 x y ....................................................................................... ............................ ……………………………………………………………… …………… ……………………………………………………………… …………… ……………………………………………………… A , B 7. 2x …………………… x = A = B x y 5 y y = 1 x y ....................................................................................... ............................ ……………………………………………………………… …………… ……………………………………………………………… …………… ………………………………………………………
8. A x y x 2y , B A = B 8 1 x y ....................................................................................... ............................ ……………………………………………………………… …………… ……………………………………………………………… …………… ……………………………………………………… …………………… ……………………………………………………… …………………… ……………………………………………………… …………………… x = 9. A A = B y2 x= 2 y 4x y , B x 2 4 y .......................................................................................
10. A A = B 3x 2 y x 4y , B x 1 7 y ....................................................................................... ............................ ……………………………………………………………… …………… ……………………………………………………………… …………… ……………………………………………………… …………………… ……………………………………………………… ……………………
8 1. A = 1 0 2 4 At 5 3 …………………………………………………………………… …………………. …………………………………………………………… …………………………. …………………………………………………………… …………………………. 2. B = 1 7 0 Bt …………………………………………………………………… …………………. …………………………………………………………… ………………………….
3. C = 2 3 0 Ct 1 5 1 7 4 5 …………………………………………………………………… …………………. …………………………………………………………… …………………………. …………………………………………………………… …………………………. 4. D = 1 2 3 4 Dt 5 6 7 8 …………………………………………………………………… …………………. …………………………………………………………… …………………………. …………………………………………………………… …………………………. 5. E = 2 3 Et 4 …………………………………………………………………… ………………….
6. G = 0 3 6 1 4 7 Gt 2 5 8 …………………………………………………………………… …………………. …………………………………………………………… …………………………. …………………………………………………………… …………………………. 7. A = 2 1 0 0 5 3 2 4 3 B = 1 4 2 3 2 0 (A + 1 5 9 B)t …………………………………………………………………… ………………….
8. A = 2 1 0 0 5 3 2 4 3 B = 1 4 2 3 2 0 At + 1 5 9 Bt …………………………………………………………………… …………………. …………………………………………………………… …………………………. …………………………………………………………… …………………………. …………………………………………………………… …………………………. …………………………………………………………… …………………………. …………………………………………………………… ………………………….
9. A = 1 2 3 0 B = 5 4 1 2 (AB)t …………………………………………………………………… …………………. …………………………………………………………… …………………………. …………………………………………………………… …………………………. …………………………………………………………… …………………………. …………………………………………………………… …………………………. …………………………………………………………… …………………………. 2 1 5 1 10. A = B = BtAt 3 0 4 2 …………………………………………………………… …………………………. …………………………………………………………… …………………………………………………………………… …………………………. …………………. …………………………………………………………… …………………………. …………………………………………………………… …………………………. ……………………………………………………………
5 1. A 3 0 4 1.1 1 5 1 , B 1 3 2 1 6 0 ,C 5 4 4 3 1 7 3A =…………………………………………………………………… ………………… 1.2 -2B =…………………………………………………………………… …………………
1.3 C =…………………………………………………………………… ………………… 1.4 4A + B =…………………………………………………………………… ………………… …………………………………………………………………… …………………… 1.5 2C - 3B …………………………………………………………………… =…………………………………………………………………… …………………… ………………… …………………………………………………………………… 3 1 2 3 1 2 …………………………………………………………………… A 4 0 8 , B 4 0 8 ……………………. 2. …………………… 1 5 4 2 5 4 …………………………………………………………………… …………………… 2.1 -2A =……………………………………. …………………………………………………………………… ……………………………………………… …………………… …………………………………….…………………………… …………………………………………………………………… …………………….. …………………….
2.2 5B =……………………………………. ……………………………………………… …………………………………….…………………………… …………………….. …………………………………….…………………………… …………………….. …………………………………….…………………………… 2.3 - A =……………………………………. …………………….. ……………………………………………… …………………………………….…………………………… …………………………………….…………………………… …………………….. …………………….. …………………………………….…………………………… …………………………………….…………………………… …………………….. …………………….. …………………………………….…………………………… 2.4 A + 3B =……………………………………. …………………….. …………………………………… …………………………………….……………………………
2.5 -4A - 2B =……………………………………. ……………………………………… …………………………………….…………………………… …………………….. …………………………………….…………………………… …………………….. …………………………………….…………………………… …………………….. …………………………………….…………………………… …………………….. …………………………………….……………………………
6 X 1. X A = 1 3 2 4 ,B = 0 3 1 5 2X = 2A + 2B .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. 0 0 .................................................... , B = 2 4 2. A = 0 0 5 3 .............................................................................................. X 3X – 3B = 2A
3. A = 3 4 2 6 X ,B = 1 4 2 6 X – A = 3B .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... ..............................................................................................
4. A = 4 3 1 2 ,B = 0 1 3 0 X -2A - X = -3B ......................................................................................... ......................................................... ......................................................................................... ......................................................... ......................................................................................... ......................................................... ......................................................................................... ......................................................... ......................................................................................... ......................................................... ......................................................................................... ......................................................... ......................................................................................... 6 4 5. A = X ......................................................... 2 0 ......................................................................................... A + X = 2A – X ......................................................... ......................................................................................... ......................................................... ................................................................................................. .................................................
A = 2 6 0 4 4 2 6. 4 2 0 2 3 5 0 2 1 ,B = 1 1 1 X A+ X = B .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. ....................................................
A = 4 1 2 1 1 1 7. 1 2 1 ,B = 1 1 1 3 2 1 2 5 1 X 2A - X = 3B .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. 3 8 1 0 8. A = ,B = 0 1 0 1 .................................................... X .............................................................................................. - (A + 3B) = X .................................................... .............................................................................................. .................................................... .............................................................................................. .............................................................................................. .................................................... .................................................... .............................................................................................. .................................................... .............................................................................................. ....................................................
A = 6 2 0 10 0 2 9. 0 4 2 ,B = 2 1 0 3 3 5 4 9 1 X 2(A + 2X) = 4B .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. 0 0 1 5 10. A = ,B = 0 0 0 2 .................................................... X .............................................................................................. -2 .................................................... ( 2A – X) = 2B + X
7.
1. A = 1 3 5 B = 2 4 AB 6 AB = …………………………………………………………………… ……….. …………………………………………………………………… ……….. …………………………………………………………………… ……….. 5 1 2 …………………………………………………………………… 2. A = B = AB ……….. 0 2 3 AB = …………………………………………………………………… …………………………………………………………………… ………. ……….. …………………………………………………………………… …………………………………………………………………… ……….. ……….. …………………………………………………………………… ………..
3. A = 1 2 3 4 B = 0 2 1 5 AB AB = …………………………………………………………………… ……….. …………………………………………………………………… ……….. …………………………………………………………………… ……….. …………………………………………………………………… ……….. 3 5 0 1 4 …………………………………………………………………… 4. A = B = AB 7 0 2 3 5 ………. AB = …………………………………………………………………… …………………………………………………………………… ……….. ……….. …………………………………………………………………… …………………………………………………………………… ………. ………..
5. A = 1 2 2 0 3 4 B = 4 5 1 2 0 1 BA BA = …………………………………………………………………… ……….. …………………………………………………………………… ……….. …………………………………………………………………… ……….. …………………………………………………………………… ………..
1 6. 4 A = B = 5 2 0 AB 1 2 AB = …………………………………………………………………… ……….. …………………………………………………………………… ……….. …………………………………………………………………… ……….. …………………………………………………………………… ……….. …………………………………………………………………… ………. 2 1 3 7 4 7. A = B = 5 0 AB 3 0 5 0 3 …………………………………………………………………… ……….. AB = …………………………………………………………………… …………………………………………………………………… ……….. ………. …………………………………………………………………… ……………………………………………………………………
9. A = 1 0 0 1 B = 5 2 4 3 AB AB = …………………………………………………………………… ……….. …………………………………………………………………… ……….. …………………………………………………………………… ……….. 1 0 5 4 10. A = B = BA …………………………………………………………………… 0 1 2 3 ……….. BA = …………………………………………………………………… ………. ……….. …………………………………………………………………… ……….. …………………………………………………………………… ………. ……….. ……………………………………………………………………
9 1. A = 1 0 2 3 ,B = 0 2 2 1 ,C = 1 1 4 5 (AB)C = A(BC) .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. ....................................................
2. – B2 A = 1 1 2 3 ,B = 0 2 1 1 A2 (A - B)(A + B) .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. ....................................................
3. A = 0 0 0 0 ,B = 1 1 2 3 ,C = 1 0 0 1
4. + B)2 A = 2 1 0 3 ,B = 1 1 2 4 (A A2+2AB + B2 .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... ..............................................................................................
5. A = 0 3 1 2 ,B = 0 1 5 2 (AB)t = BtAt .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. ....................................................
6. A = 1 0 2 5 , B = 1 2 3 4 (A + B)C = AC + BC , C = 0 1 4 0
7. A = 2 1 3 0 (At)t = A .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... 3 8 .............................................................................................. 8. A = (3A)t = 3At 0 1 .................................................... .............................................................................................. .................................................... .............................................................................................. ..............................................................................................
9. A = 6 1 7 0 ,I = 1 0 0 1 IA = A .............................................................................................. .................................................... 5 0 0 1 10. A = ,B = 0 0 0 2 .............................................................................................. X .................................................... .............................................................................................. -2 ( 2A – X) = 2B + X .................................................... .............................................................................................. ...................................................................................... .................................................... ............................................................ .............................................................................................. ...................................................................................... .................................................... ............................................................ 1 2 6 2 .............................................................................................. 10. A = , B = ......................................................................................
10 1
1. A = 1 2 0 3 A ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… 1 2 2. A = A 1 0 ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… …………………………………………………………………
3. A = 2 1 1 A 2 ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… 3 1 4. A = A ………………………………………………………………… 1 2 …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… ……………………………
5. A = 1 2 3 4 A ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… 4 3 6. A = A 1 2 …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… ………………………………………………………………… …………………………… ……………………………
2 B A 2 1. A = 3 2 1 4 , B = 4 2 1 3 ………………………………………………………………… …………………………… …………………………………………………………………
A = 1 1 0 2 3 1 2. 2 1 1 , B = 2 0 2 7 1 5 1 1 1 ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… …………………………………………………………………
3 3A - 3B = A-1 1 1 B-1 A + B = 3 1 1 2 1 0 ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… …………………………………………………………………
4 2A-1 - B-1 = A B A-1 + B-1 = 2 2 1 3 1 1 4 3 ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… ……………………………
11
1 1. 6 ……………………………………………………………….. 2. 3 ……………………………………………………………….. 3. 17 ……………………………………………………………….. 4. 0 ……………………………………………………………….. 5. 23 ………………………………………………………………..
2 1. 0 2 3 4 ……………………………………………………………….. 2. 2 7 3 5 ……………………………………………………………….. 3. 5 3 2 3 ……………………………………………………………….. 4. 7 0 6 1 ………………………………………………………………..
5. 3 4 2 5 ……………………………………………………………….. 3 1. 1 2 3 1 3 4 2 4 5 =…………………………………………………………… ………… =…………………………………………………………… ………… =…………………………………………………………… ………… =…………………………………………………………… …………
3 1 2 1 2 0 2. 0 3 5 =…………………………………………………………… ………… =…………………………………………………………… ………… =…………………………………………………………… ………… =…………………………………………………………… ………… ……………………………………………………………….. 3. 2 1 4 1 0 2 5 3 1 =……………………………………………………… ……………… =……………………………………………………… ………………
1 0 5 2 0 8 4. 2 3 7 =…………………………………………………………… ………… =…………………………………………………………… ………… =…………………………………………………………… ………… =…………………………………………………………… ………… 5 3 2 …………………………………………………………… 4 0 1 5. ….. 2 4 2 =…………………………………………………………… ………… =…………………………………………………………… ………… =…………………………………………………………… ………… =…………………………………………………………… …………
12 1. A = 2 0 5 3 1 3 det (A) = 4 2 2 det(At) .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... ..............................................................................................
2. A = 2 0 0 8 1 0 det (A) = 4 9 2 a11a22a33 .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... ..............................................................................................
3. A = 2 3 1 0 ,B = 8 12 1 0 det(B) = 4 det(A) .............................................................................................. .................................................... .............................................................................................. .................................................... ..............................................................................................
4. A = 5 4 2 2 ,B = 10 8 4 4 det(B) = 22 det(A) .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... 1 0 0
6. A = 3 5 4 6 det(A-1) = .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. ....................................................
7. A = 1 6 2 3 ,B = 4 8 3 5 det(AB) = det(A)det(B) .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... ..............................................................................................
8. A = 0 2 1 7 det(A3) = det(A)3 .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... ..............................................................................................
9. A = 1 2 3 3 4 2 5 6 0 det(A) = det(B) , B = 1 2 3 6 10 11 5 6 0
A = 2 3 5 7 1 0 10. 2 4 1 , B = 2 4 2 7 5 3 1 0 1 det(A) = - det(B) .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. ....................................................
13 1. A = 4 5 0 2 M11(A)
2. B = 3. C = 4. A = 1 2 M21(B) 3 7 8 0 M12(C) 5 1 1 2 4 3 1 5 0 1 2 M12(A)
5. B = 1 0 2 5 3 4 M23(B) 1 2 5 6. C = 2 0 1 1 2 4 3 6 0 M33(C)
8. A = B = 4 3 1 5 0 1 7. 1 2 2 1 0 2 5 3 4 M13(A) M31(B) 1 2 5 9. C = 2 0 1 1 2 4 3 6 0 M13(C)
1 2 3 4 10. A = 0 1 5 7 1 2 6 0 2 4 3 8 M34(A)
14 1. A = 4 5 0 2 C12(A)
2. B = 3. C = 1 2 3 7 8 0 5 1 C22(B) C21(C)
5. A = B = 4 3 1 5 0 1 4. 1 2 2 1 0 2 5 3 4 C13(A) C33(B) 1 2 5 6. C = 2 0 1 1 2 4 3 6 0 C21(C)
8. A = B = 4 3 1 5 0 1 7. 1 2 2 1 0 2 5 3 4 1 2 5 C22(A) C11(B)
9. C = 2 0 1 1 2 4 C31(C) 3 6 0 1 2 3 4 10. A = 0 1 5 7 1 2 6 0 2 4 3 8 C14(A)
15 1. A = 1 2 2. B = 5 4 3 2 3. C = 8 0 det(A) 3 4 5 1 det(B) det(C)
5. A = B = 4 3 1 5 0 1 4. 1 2 2 1 0 0 5 3 4 1 2 5 det(A) det(B)
6. C = 2 0 1 1 2 4 3 6 0 det(C)
A = 4 3 1 5 0 1 7. 0 2 2 det(A) 2 3 7 8. B = 0 1 2 6 1 0 0 0 4 4 1 3 0 det(B)
0 9. C = 2 1 3 5 1 2 4 0 3 5 1 1 2 3 4 det(C)
1 0 3 0 10. A = 0 1 5 7 1 2 6 0 2 4 3 8 det(A)
16 1. A = 6 8 2 3 adj (A) ……………………………………………………………………… ……………………. ……………………………………………………………………… ……………………. ………………………………………………………………………
2. B = 4 3 5 2 adj (B) ……………………………………………………………………… ……………………. ……………………………………………………………………… ……………………. ……………………………………………………………………… ……………………. ……………………………………………………………………… ……………………. ………………………………………………………………… 0 4 1 2 …………………………. 3. C = adj (C) ……………………………………………………………………… ……………………………………………………………………… ……………………. ……………………. ………………………………………………………………… ………………………….
4. D = 3 6 5 4 adj (D) ……………………………………………………………………… ……………………. ……………………………………………………………………… ……………………. ……………………………………………………………………… ……………………. ……………………………………………………………………… ……………………. ………………………………………………………………… ………………………….
A = 2 1 4 5 6 2 5. 3 3 1 adj (A) ……………………………………………………………………… ……………………. ……………………………………………………………………… ……………………. ……………………………………………………………………… ……………………. ……………………………………………………………………… ……………………. ………………………………………………………………… …………………………. 3 4 2 6 3 1 6. B = adj (B) ……………………………………………………………………… 4 7 8 ……………………. ………………………………………………………………… ……………………………………………………………………… …………………………. ……………………. ……………………………………………………………………… ………………………………………………………………………
A = 1 3 1 12 4 1 7. 5 3 6 adj (A) ……………………………………………………………………… ……………………. ……………………………………………………………………… ……………………. ……………………………………………………………………… ……………………. ……………………………………………………………………… ……………………. …………………………………………………………………
8. B = 10 5 30 0 10 0 10 5 adj (B) 1 ……………………………………………………………………… ……………………. ……………………………………………………………………… ……………………. ……………………………………………………………………… ……………………. ……………………………………………………………………… ……………………. ………………………………………………………………… …………………………. 7 3 3 ……………………………………………………………………… 1 1 0 9. A = adj (A) ……………………. 1 0 1 ………………………………………………………………… …………………………. ……………………………………………………………………… ……………………. ……………………………………………………………………… ……………………. ……………………………………………………………………… …………………………………………………………………
4 8 8 4 10. B = 1 2 0 2 0 1 2 3 adj (B) 2 4 1 4 ……………………………………………………………………… ……………………. ……………………………………………………………………… ……………………. ……………………………………………………………………… ……………………. ……………………………………………………………………… ……………………. ………………………………………………………………… ………………………….
17 1. A = 1 0 5 3 A-1 ( (1 ………………………………………………………………… ……………………. ………………………………………………………………… …………………….. ………………………………………………………………… ……………………. ………………………………………………………………… …………………….. ………………………………………………………………… ……………………. …………………………………………………………………
2. A = 1 5 2 6 A-1 ( (1 ………………………………………………………………… ……………………. ………………………………………………………………… …………………….. ………………………………………………………………… ……………………. ………………………………………………………………… …………………….. ………………………………………………………………… …………………….
3. A = 3 2 4 6 A-1 ( (1 ………………………………………………………………… ……………………. ………………………………………………………………… ……………………..
4. (1 A = 6 8 2 4 A-1 (
A = 2 3 1 1 0 0 5. 2 1 4 A-1 ( (1 ………………………………………………………………… ……………………. ………………………………………………………………… …………………….. ………………………………………………………………… ……………………. ………………………………………………………………… …………………….. ………………………………………………………………… ……………………. …………………………………………………………………
A = 17 11 1 11 7 0 6. 2 3 2 A-1 ( (1 ………………………………………………………………… ……………………. ………………………………………………………………… …………………….. …………………………………………………………………
(1 A = 4 2 3 1 3 4 7. 1 1 3 A-1 (
A = 0 2 1 2 0 1 8. 1 0 3 A-1 ( (1 ………………………………………………………………… ……………………. ………………………………………………………………… …………………….. ………………………………………………………………… ……………………. ………………………………………………………………… …………………….. ………………………………………………………………… ……………………. ………………………………………………………………… ……………………..
1 ( A = 0 0 1 1 0 0 2 1 1 0 3 9. 0 2 1 1 A-1 (1 ………………………………………………………………… ……………………. ………………………………………………………………… …………………….. ………………………………………………………………… …………………….
2 ( A = (1 1 1 3 0 0 1 1 1 2 1 4 10. 0 1 1 0 A-1
18 1. 2x - 3y = -1 10x + y = 11 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… ……………………………………………………………………
2. 3x - 6y = 2 5x + 4y = 1 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ………………………
3. 4x - 7y = -2 - x + 2y = 1 …………………………………………………………………… ………………………
4. 8x + 6y = -6 -2x + 4y = -26 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ………………………
5. x + 3y = 4 -2x - 4y = -6 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ………………………
6. x + y - z = -1 4x - 3y + 2z = 16 2x - 2y - 3z = 5 …………………………………………………………………… ……………………… …………………………………………………………………… ………………………
7. 6x + y + 20z = 27
8. 4x + y = 6 2x - y + 4z = 12 -3x + 2y + z = 4 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… ……………………………………………………………………
9. x + y + z = 5 2x - y + 2z = 4 3x + 2y - z = -3 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ………………………
10. x + 2y = 0 z x + 3y = -4 + w = 4
19 1. 2x - 3y = -1 10x + y = 11 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… ……………………………………………………………………
2. 3x - 6y = 2 5x + 4y = 1 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… ……………………………………………………………………
3. 4x - 7y = -2 - x + 2y = 1 ……………………………………………………………………
4. 8x + 6y = -6 -2x + 4y = -26 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ………………………
5. x + 3y = 4 -2x - 4y = -6 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… ……………………………………………………………………
6. x + y - z = -1 4x - 3y + 2z = 16 2x - 2y - 3z = 5 …………………………………………………………………… ………………………
7. 6x + y + 20z x - y = 27 = 0 y + 3z = 4 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… ……………………………………………………………………
8. 4x + y = 6 2x - y + 4z = 12 -3x + 2y + z = 4 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ………………………
9. x + y + z = 5 2x - y + 2z = 4 3x + 2y - z = -3 …………………………………………………………………… ……………………… …………………………………………………………………… ………………………
10. x + 2y = 0
20 1. x + y - z = -1 4x - 3y + 2z = 16 2x - 2y - 3z = 5 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ………………………
2. 2x + 4y - 2z = 0 3x + 5y = 1 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… ……………………………………………………………………
3. 6x + y x - y + 20z = 27 = 0 y + 3z = 4
4. x + 2y + z= 8 3x + 7y + 6z = 26 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ………………………
5. 4x + y = 6 2x - y + 4z = 12 -3x + 2y + z = 4 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ………………………
6. x + y + z = 5 2x - y + 2z = 4 3x + 2y - z = -3
7. x + 2y = 0 z = -4 x + 3y + 4x w = 4 + 2w = -2 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… ……………………………………………………………………
8. x - 3z = -2 3x + y - 2z = 2x + 2y + z = 4 5 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… ……………………………………………………………………
9. x + 5y + 2z = 6 2x + 10y + 2z x + 5y + z = 6 = 3 4x + 20y + 5z = 15 …………………………………………………………………… ………………………
10. 4w + x + 2y + 2z= 11 12w + 3x + 6y + 5z = 30 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… ……………………………………………………………………
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  • 2. 40 1. - 1. x + 3y x – 2y = = 8 3 …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. ……………………………………………………………………
  • 3. 2. x- y = 2x + 3y = -4 22 …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. 3. x+ y+z = 6 …………………………………………………………………… x–y+z = 2 ……………………………. = x+y-z 0
  • 4.
  • 5. 4. x + 2y -z = 3 3x + y = 6 2x + y = 1 …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. ……………………………………………………………………
  • 6. 5. 2x - 3y + z = 8 - x + 4y + 2z = -4 3x - y + 2z = 9 …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… …………………………….
  • 7. 6. x - y + 2z = -3 y – 3z = 5 x + 4y – 8z = 17 …………………………………………………………………… ……………………………. …………………………………………………………………… …………………………….
  • 8.
  • 9. 7. x + y = 2 x + 3y + z = 5 3x + y - z = 3 …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… …………………………….
  • 10. 8. 2x + 2y + 3z + 2t = -3 x + y + 2z + 2t = 6 2y + 5z + 2t = 5 x + y + 3z + 4t = 1 …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. …………………………………………………………………… ……………………………. ……………………………………………………………………
  • 11. 2 1 1. A = B = 3 5 9 3. C = 2 5 0 1 2. 1 1 3 1 3 5 3 4
  • 12. 2 2
  • 14. 3 1. 1. 4 1 7 9 …………………….. 3 1 1 3 2. 0 2 2 1 0 0 3 0 0 0 0 6 …………………….. 3. 2 3 4 …………………….. 4. 2 0 0 2 …………………….. 5. 0 0 0 0 0 0 ……………………..
  • 15. 2. 1) 2 3 2 2) 0 1 4 2 5 3) 1 0 0 1 9) 6 2 0 3 0 0 1 4) 0 0 0 0 5) 0 1 1 0 6) 1 0 0 0 1 0 0 0 1 7) 0 0 0 8) 0 0 0 0 0 5 12) 0 1 8 . 10) 5 0 1 2 14) 0 0 0 15) 9 0 0 8 5 0 3 0 2 0 0 0 2 3 13) 2 0 0 1 5 11) 0 .
  • 17. 1. A 1 2 2x 6 1 , B A = B y 8 3 2 x y ....................................................................................... ............................ ……………………………………………………………… …………… ……………………………………………………………… 3x y 2 1 2 2. …………… A x = A = B 2y , B 5 y = 10 x 5 y ....................................................................................... ............................ ……………………………………………………………… …………… ……………………………………………………………… 3. …………… x = A = B A o 4 3 8 y = , B x x x 3 y 8 y x = …………… y = …………… .......................................................................................
  • 18. 3 4. A 1 4 2 0 1 x 4 0 8 2 5 8 , B x 5 A = B 3 4 4 x y y ....................................................................................... ............................ ……………………………………………………………… …………… ……………………………………………………………… 1 9 2 3 3x 2 x y 5. …………… 3 4 5 2 5 x = A 4 y = A = B , B 3 4 5 2 5 4 x y ....................................................................................... ............................ ……………………………………………………………… ……………
  • 19. 6. A x y x y A = B 1 , B 7 x y ....................................................................................... ............................ ……………………………………………………………… …………… ……………………………………………………………… …………… ……………………………………………………… A , B 7. 2x …………………… x = A = B x y 5 y y = 1 x y ....................................................................................... ............................ ……………………………………………………………… …………… ……………………………………………………………… …………… ………………………………………………………
  • 20. 8. A x y x 2y , B A = B 8 1 x y ....................................................................................... ............................ ……………………………………………………………… …………… ……………………………………………………………… …………… ……………………………………………………… …………………… ……………………………………………………… …………………… ……………………………………………………… …………………… x = 9. A A = B y2 x= 2 y 4x y , B x 2 4 y .......................................................................................
  • 21. 10. A A = B 3x 2 y x 4y , B x 1 7 y ....................................................................................... ............................ ……………………………………………………………… …………… ……………………………………………………………… …………… ……………………………………………………… …………………… ……………………………………………………… ……………………
  • 22. 8 1. A = 1 0 2 4 At 5 3 …………………………………………………………………… …………………. …………………………………………………………… …………………………. …………………………………………………………… …………………………. 2. B = 1 7 0 Bt …………………………………………………………………… …………………. …………………………………………………………… ………………………….
  • 23. 3. C = 2 3 0 Ct 1 5 1 7 4 5 …………………………………………………………………… …………………. …………………………………………………………… …………………………. …………………………………………………………… …………………………. 4. D = 1 2 3 4 Dt 5 6 7 8 …………………………………………………………………… …………………. …………………………………………………………… …………………………. …………………………………………………………… …………………………. 5. E = 2 3 Et 4 …………………………………………………………………… ………………….
  • 24. 6. G = 0 3 6 1 4 7 Gt 2 5 8 …………………………………………………………………… …………………. …………………………………………………………… …………………………. …………………………………………………………… …………………………. 7. A = 2 1 0 0 5 3 2 4 3 B = 1 4 2 3 2 0 (A + 1 5 9 B)t …………………………………………………………………… ………………….
  • 25. 8. A = 2 1 0 0 5 3 2 4 3 B = 1 4 2 3 2 0 At + 1 5 9 Bt …………………………………………………………………… …………………. …………………………………………………………… …………………………. …………………………………………………………… …………………………. …………………………………………………………… …………………………. …………………………………………………………… …………………………. …………………………………………………………… ………………………….
  • 26. 9. A = 1 2 3 0 B = 5 4 1 2 (AB)t …………………………………………………………………… …………………. …………………………………………………………… …………………………. …………………………………………………………… …………………………. …………………………………………………………… …………………………. …………………………………………………………… …………………………. …………………………………………………………… …………………………. 2 1 5 1 10. A = B = BtAt 3 0 4 2 …………………………………………………………… …………………………. …………………………………………………………… …………………………………………………………………… …………………………. …………………. …………………………………………………………… …………………………. …………………………………………………………… …………………………. ……………………………………………………………
  • 27. 5 1. A 3 0 4 1.1 1 5 1 , B 1 3 2 1 6 0 ,C 5 4 4 3 1 7 3A =…………………………………………………………………… ………………… 1.2 -2B =…………………………………………………………………… …………………
  • 28. 1.3 C =…………………………………………………………………… ………………… 1.4 4A + B =…………………………………………………………………… ………………… …………………………………………………………………… …………………… 1.5 2C - 3B …………………………………………………………………… =…………………………………………………………………… …………………… ………………… …………………………………………………………………… 3 1 2 3 1 2 …………………………………………………………………… A 4 0 8 , B 4 0 8 ……………………. 2. …………………… 1 5 4 2 5 4 …………………………………………………………………… …………………… 2.1 -2A =……………………………………. …………………………………………………………………… ……………………………………………… …………………… …………………………………….…………………………… …………………………………………………………………… …………………….. …………………….
  • 29. 2.2 5B =……………………………………. ……………………………………………… …………………………………….…………………………… …………………….. …………………………………….…………………………… …………………….. …………………………………….…………………………… 2.3 - A =……………………………………. …………………….. ……………………………………………… …………………………………….…………………………… …………………………………….…………………………… …………………….. …………………….. …………………………………….…………………………… …………………………………….…………………………… …………………….. …………………….. …………………………………….…………………………… 2.4 A + 3B =……………………………………. …………………….. …………………………………… …………………………………….……………………………
  • 30. 2.5 -4A - 2B =……………………………………. ……………………………………… …………………………………….…………………………… …………………….. …………………………………….…………………………… …………………….. …………………………………….…………………………… …………………….. …………………………………….…………………………… …………………….. …………………………………….……………………………
  • 31. 6 X 1. X A = 1 3 2 4 ,B = 0 3 1 5 2X = 2A + 2B .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. 0 0 .................................................... , B = 2 4 2. A = 0 0 5 3 .............................................................................................. X 3X – 3B = 2A
  • 32. 3. A = 3 4 2 6 X ,B = 1 4 2 6 X – A = 3B .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... ..............................................................................................
  • 33. 4. A = 4 3 1 2 ,B = 0 1 3 0 X -2A - X = -3B ......................................................................................... ......................................................... ......................................................................................... ......................................................... ......................................................................................... ......................................................... ......................................................................................... ......................................................... ......................................................................................... ......................................................... ......................................................................................... ......................................................... ......................................................................................... 6 4 5. A = X ......................................................... 2 0 ......................................................................................... A + X = 2A – X ......................................................... ......................................................................................... ......................................................... ................................................................................................. .................................................
  • 34. A = 2 6 0 4 4 2 6. 4 2 0 2 3 5 0 2 1 ,B = 1 1 1 X A+ X = B .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. ....................................................
  • 35. A = 4 1 2 1 1 1 7. 1 2 1 ,B = 1 1 1 3 2 1 2 5 1 X 2A - X = 3B .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. 3 8 1 0 8. A = ,B = 0 1 0 1 .................................................... X .............................................................................................. - (A + 3B) = X .................................................... .............................................................................................. .................................................... .............................................................................................. .............................................................................................. .................................................... .................................................... .............................................................................................. .................................................... .............................................................................................. ....................................................
  • 36. A = 6 2 0 10 0 2 9. 0 4 2 ,B = 2 1 0 3 3 5 4 9 1 X 2(A + 2X) = 4B .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. 0 0 1 5 10. A = ,B = 0 0 0 2 .................................................... X .............................................................................................. -2 .................................................... ( 2A – X) = 2B + X
  • 37. 7.
  • 38. 1. A = 1 3 5 B = 2 4 AB 6 AB = …………………………………………………………………… ……….. …………………………………………………………………… ……….. …………………………………………………………………… ……….. 5 1 2 …………………………………………………………………… 2. A = B = AB ……….. 0 2 3 AB = …………………………………………………………………… …………………………………………………………………… ………. ……….. …………………………………………………………………… …………………………………………………………………… ……….. ……….. …………………………………………………………………… ………..
  • 39. 3. A = 1 2 3 4 B = 0 2 1 5 AB AB = …………………………………………………………………… ……….. …………………………………………………………………… ……….. …………………………………………………………………… ……….. …………………………………………………………………… ……….. 3 5 0 1 4 …………………………………………………………………… 4. A = B = AB 7 0 2 3 5 ………. AB = …………………………………………………………………… …………………………………………………………………… ……….. ……….. …………………………………………………………………… …………………………………………………………………… ………. ………..
  • 40. 5. A = 1 2 2 0 3 4 B = 4 5 1 2 0 1 BA BA = …………………………………………………………………… ……….. …………………………………………………………………… ……….. …………………………………………………………………… ……….. …………………………………………………………………… ………..
  • 41. 1 6. 4 A = B = 5 2 0 AB 1 2 AB = …………………………………………………………………… ……….. …………………………………………………………………… ……….. …………………………………………………………………… ……….. …………………………………………………………………… ……….. …………………………………………………………………… ………. 2 1 3 7 4 7. A = B = 5 0 AB 3 0 5 0 3 …………………………………………………………………… ……….. AB = …………………………………………………………………… …………………………………………………………………… ……….. ………. …………………………………………………………………… ……………………………………………………………………
  • 42.
  • 43. 9. A = 1 0 0 1 B = 5 2 4 3 AB AB = …………………………………………………………………… ……….. …………………………………………………………………… ……….. …………………………………………………………………… ……….. 1 0 5 4 10. A = B = BA …………………………………………………………………… 0 1 2 3 ……….. BA = …………………………………………………………………… ………. ……….. …………………………………………………………………… ……….. …………………………………………………………………… ………. ……….. ……………………………………………………………………
  • 44. 9 1. A = 1 0 2 3 ,B = 0 2 2 1 ,C = 1 1 4 5 (AB)C = A(BC) .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. ....................................................
  • 45. 2. – B2 A = 1 1 2 3 ,B = 0 2 1 1 A2 (A - B)(A + B) .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. ....................................................
  • 46. 3. A = 0 0 0 0 ,B = 1 1 2 3 ,C = 1 0 0 1
  • 47.
  • 48. 4. + B)2 A = 2 1 0 3 ,B = 1 1 2 4 (A A2+2AB + B2 .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... ..............................................................................................
  • 49. 5. A = 0 3 1 2 ,B = 0 1 5 2 (AB)t = BtAt .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. ....................................................
  • 50. 6. A = 1 0 2 5 , B = 1 2 3 4 (A + B)C = AC + BC , C = 0 1 4 0
  • 51.
  • 52. 7. A = 2 1 3 0 (At)t = A .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... 3 8 .............................................................................................. 8. A = (3A)t = 3At 0 1 .................................................... .............................................................................................. .................................................... .............................................................................................. ..............................................................................................
  • 53. 9. A = 6 1 7 0 ,I = 1 0 0 1 IA = A .............................................................................................. .................................................... 5 0 0 1 10. A = ,B = 0 0 0 2 .............................................................................................. X .................................................... .............................................................................................. -2 ( 2A – X) = 2B + X .................................................... .............................................................................................. ...................................................................................... .................................................... ............................................................ .............................................................................................. ...................................................................................... .................................................... ............................................................ 1 2 6 2 .............................................................................................. 10. A = , B = ......................................................................................
  • 54. 10 1
  • 55. 1. A = 1 2 0 3 A ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… 1 2 2. A = A 1 0 ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… …………………………………………………………………
  • 56. 3. A = 2 1 1 A 2 ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… 3 1 4. A = A ………………………………………………………………… 1 2 …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… ……………………………
  • 57. 5. A = 1 2 3 4 A ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… 4 3 6. A = A 1 2 …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… ………………………………………………………………… …………………………… ……………………………
  • 58. 2 B A 2 1. A = 3 2 1 4 , B = 4 2 1 3 ………………………………………………………………… …………………………… …………………………………………………………………
  • 59.
  • 60. A = 1 1 0 2 3 1 2. 2 1 1 , B = 2 0 2 7 1 5 1 1 1 ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… …………………………………………………………………
  • 61. 3 3A - 3B = A-1 1 1 B-1 A + B = 3 1 1 2 1 0 ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… …………………………………………………………………
  • 62. 4 2A-1 - B-1 = A B A-1 + B-1 = 2 2 1 3 1 1 4 3 ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… …………………………… ………………………………………………………………… ……………………………
  • 63. 11
  • 65. 2 1. 0 2 3 4 ……………………………………………………………….. 2. 2 7 3 5 ……………………………………………………………….. 3. 5 3 2 3 ……………………………………………………………….. 4. 7 0 6 1 ………………………………………………………………..
  • 66. 5. 3 4 2 5 ……………………………………………………………….. 3 1. 1 2 3 1 3 4 2 4 5 =…………………………………………………………… ………… =…………………………………………………………… ………… =…………………………………………………………… ………… =…………………………………………………………… …………
  • 67. 3 1 2 1 2 0 2. 0 3 5 =…………………………………………………………… ………… =…………………………………………………………… ………… =…………………………………………………………… ………… =…………………………………………………………… ………… ……………………………………………………………….. 3. 2 1 4 1 0 2 5 3 1 =……………………………………………………… ……………… =……………………………………………………… ………………
  • 68. 1 0 5 2 0 8 4. 2 3 7 =…………………………………………………………… ………… =…………………………………………………………… ………… =…………………………………………………………… ………… =…………………………………………………………… ………… 5 3 2 …………………………………………………………… 4 0 1 5. ….. 2 4 2 =…………………………………………………………… ………… =…………………………………………………………… ………… =…………………………………………………………… ………… =…………………………………………………………… …………
  • 69. 12 1. A = 2 0 5 3 1 3 det (A) = 4 2 2 det(At) .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... ..............................................................................................
  • 70. 2. A = 2 0 0 8 1 0 det (A) = 4 9 2 a11a22a33 .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... ..............................................................................................
  • 71. 3. A = 2 3 1 0 ,B = 8 12 1 0 det(B) = 4 det(A) .............................................................................................. .................................................... .............................................................................................. .................................................... ..............................................................................................
  • 72. 4. A = 5 4 2 2 ,B = 10 8 4 4 det(B) = 22 det(A) .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... 1 0 0
  • 73. 6. A = 3 5 4 6 det(A-1) = .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. ....................................................
  • 74. 7. A = 1 6 2 3 ,B = 4 8 3 5 det(AB) = det(A)det(B) .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... ..............................................................................................
  • 75. 8. A = 0 2 1 7 det(A3) = det(A)3 .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... ..............................................................................................
  • 76. 9. A = 1 2 3 3 4 2 5 6 0 det(A) = det(B) , B = 1 2 3 6 10 11 5 6 0
  • 77.
  • 78. A = 2 3 5 7 1 0 10. 2 4 1 , B = 2 4 2 7 5 3 1 0 1 det(A) = - det(B) .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. .................................................... .............................................................................................. ....................................................
  • 79. 13 1. A = 4 5 0 2 M11(A)
  • 80. 2. B = 3. C = 4. A = 1 2 M21(B) 3 7 8 0 M12(C) 5 1 1 2 4 3 1 5 0 1 2 M12(A)
  • 81. 5. B = 1 0 2 5 3 4 M23(B) 1 2 5 6. C = 2 0 1 1 2 4 3 6 0 M33(C)
  • 82. 8. A = B = 4 3 1 5 0 1 7. 1 2 2 1 0 2 5 3 4 M13(A) M31(B) 1 2 5 9. C = 2 0 1 1 2 4 3 6 0 M13(C)
  • 83. 1 2 3 4 10. A = 0 1 5 7 1 2 6 0 2 4 3 8 M34(A)
  • 84. 14 1. A = 4 5 0 2 C12(A)
  • 85. 2. B = 3. C = 1 2 3 7 8 0 5 1 C22(B) C21(C)
  • 86. 5. A = B = 4 3 1 5 0 1 4. 1 2 2 1 0 2 5 3 4 C13(A) C33(B) 1 2 5 6. C = 2 0 1 1 2 4 3 6 0 C21(C)
  • 87. 8. A = B = 4 3 1 5 0 1 7. 1 2 2 1 0 2 5 3 4 1 2 5 C22(A) C11(B)
  • 88. 9. C = 2 0 1 1 2 4 C31(C) 3 6 0 1 2 3 4 10. A = 0 1 5 7 1 2 6 0 2 4 3 8 C14(A)
  • 89.
  • 90. 15 1. A = 1 2 2. B = 5 4 3 2 3. C = 8 0 det(A) 3 4 5 1 det(B) det(C)
  • 91. 5. A = B = 4 3 1 5 0 1 4. 1 2 2 1 0 0 5 3 4 1 2 5 det(A) det(B)
  • 92. 6. C = 2 0 1 1 2 4 3 6 0 det(C)
  • 93. A = 4 3 1 5 0 1 7. 0 2 2 det(A) 2 3 7 8. B = 0 1 2 6 1 0 0 0 4 4 1 3 0 det(B)
  • 94. 0 9. C = 2 1 3 5 1 2 4 0 3 5 1 1 2 3 4 det(C)
  • 95. 1 0 3 0 10. A = 0 1 5 7 1 2 6 0 2 4 3 8 det(A)
  • 97. 2. B = 4 3 5 2 adj (B) ……………………………………………………………………… ……………………. ……………………………………………………………………… ……………………. ……………………………………………………………………… ……………………. ……………………………………………………………………… ……………………. ………………………………………………………………… 0 4 1 2 …………………………. 3. C = adj (C) ……………………………………………………………………… ……………………………………………………………………… ……………………. ……………………. ………………………………………………………………… ………………………….
  • 99. A = 2 1 4 5 6 2 5. 3 3 1 adj (A) ……………………………………………………………………… ……………………. ……………………………………………………………………… ……………………. ……………………………………………………………………… ……………………. ……………………………………………………………………… ……………………. ………………………………………………………………… …………………………. 3 4 2 6 3 1 6. B = adj (B) ……………………………………………………………………… 4 7 8 ……………………. ………………………………………………………………… ……………………………………………………………………… …………………………. ……………………. ……………………………………………………………………… ………………………………………………………………………
  • 100. A = 1 3 1 12 4 1 7. 5 3 6 adj (A) ……………………………………………………………………… ……………………. ……………………………………………………………………… ……………………. ……………………………………………………………………… ……………………. ……………………………………………………………………… ……………………. …………………………………………………………………
  • 101. 8. B = 10 5 30 0 10 0 10 5 adj (B) 1 ……………………………………………………………………… ……………………. ……………………………………………………………………… ……………………. ……………………………………………………………………… ……………………. ……………………………………………………………………… ……………………. ………………………………………………………………… …………………………. 7 3 3 ……………………………………………………………………… 1 1 0 9. A = adj (A) ……………………. 1 0 1 ………………………………………………………………… …………………………. ……………………………………………………………………… ……………………. ……………………………………………………………………… ……………………. ……………………………………………………………………… …………………………………………………………………
  • 102. 4 8 8 4 10. B = 1 2 0 2 0 1 2 3 adj (B) 2 4 1 4 ……………………………………………………………………… ……………………. ……………………………………………………………………… ……………………. ……………………………………………………………………… ……………………. ……………………………………………………………………… ……………………. ………………………………………………………………… ………………………….
  • 103. 17 1. A = 1 0 5 3 A-1 ( (1 ………………………………………………………………… ……………………. ………………………………………………………………… …………………….. ………………………………………………………………… ……………………. ………………………………………………………………… …………………….. ………………………………………………………………… ……………………. …………………………………………………………………
  • 104. 2. A = 1 5 2 6 A-1 ( (1 ………………………………………………………………… ……………………. ………………………………………………………………… …………………….. ………………………………………………………………… ……………………. ………………………………………………………………… …………………….. ………………………………………………………………… …………………….
  • 106. 4. (1 A = 6 8 2 4 A-1 (
  • 107.
  • 110. (1 A = 4 2 3 1 3 4 7. 1 1 3 A-1 (
  • 111.
  • 115.
  • 116. 18 1. 2x - 3y = -1 10x + y = 11 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… ……………………………………………………………………
  • 117. 2. 3x - 6y = 2 5x + 4y = 1 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ………………………
  • 118. 3. 4x - 7y = -2 - x + 2y = 1 …………………………………………………………………… ………………………
  • 119.
  • 120. 4. 8x + 6y = -6 -2x + 4y = -26 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ………………………
  • 121. 5. x + 3y = 4 -2x - 4y = -6 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ………………………
  • 122. 6. x + y - z = -1 4x - 3y + 2z = 16 2x - 2y - 3z = 5 …………………………………………………………………… ……………………… …………………………………………………………………… ………………………
  • 123. 7. 6x + y + 20z = 27
  • 124.
  • 125. 8. 4x + y = 6 2x - y + 4z = 12 -3x + 2y + z = 4 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… ……………………………………………………………………
  • 126. 9. x + y + z = 5 2x - y + 2z = 4 3x + 2y - z = -3 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ………………………
  • 127. 10. x + 2y = 0 z x + 3y = -4 + w = 4
  • 128.
  • 129. 19 1. 2x - 3y = -1 10x + y = 11 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… ……………………………………………………………………
  • 130. 2. 3x - 6y = 2 5x + 4y = 1 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… ……………………………………………………………………
  • 131. 3. 4x - 7y = -2 - x + 2y = 1 ……………………………………………………………………
  • 132.
  • 133. 4. 8x + 6y = -6 -2x + 4y = -26 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ………………………
  • 134. 5. x + 3y = 4 -2x - 4y = -6 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… ……………………………………………………………………
  • 135. 6. x + y - z = -1 4x - 3y + 2z = 16 2x - 2y - 3z = 5 …………………………………………………………………… ………………………
  • 136.
  • 137. 7. 6x + y + 20z x - y = 27 = 0 y + 3z = 4 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… ……………………………………………………………………
  • 138. 8. 4x + y = 6 2x - y + 4z = 12 -3x + 2y + z = 4 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ………………………
  • 139. 9. x + y + z = 5 2x - y + 2z = 4 3x + 2y - z = -3 …………………………………………………………………… ……………………… …………………………………………………………………… ………………………
  • 141.
  • 142. 20 1. x + y - z = -1 4x - 3y + 2z = 16 2x - 2y - 3z = 5 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ………………………
  • 143. 2. 2x + 4y - 2z = 0 3x + 5y = 1 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… ……………………………………………………………………
  • 144. 3. 6x + y x - y + 20z = 27 = 0 y + 3z = 4
  • 145.
  • 146. 4. x + 2y + z= 8 3x + 7y + 6z = 26 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ………………………
  • 147. 5. 4x + y = 6 2x - y + 4z = 12 -3x + 2y + z = 4 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ………………………
  • 148. 6. x + y + z = 5 2x - y + 2z = 4 3x + 2y - z = -3
  • 149.
  • 150. 7. x + 2y = 0 z = -4 x + 3y + 4x w = 4 + 2w = -2 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… ……………………………………………………………………
  • 151. 8. x - 3z = -2 3x + y - 2z = 2x + 2y + z = 4 5 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… ……………………………………………………………………
  • 152. 9. x + 5y + 2z = 6 2x + 10y + 2z x + 5y + z = 6 = 3 4x + 20y + 5z = 15 …………………………………………………………………… ………………………
  • 153.
  • 154. 10. 4w + x + 2y + 2z= 11 12w + 3x + 6y + 5z = 30 …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… …………………………………………………………………… ……………………… ……………………………………………………………………