The document discusses matrices and matrix operations. It defines what a matrix is and introduces common matrix terminology like elements, rows, columns, and dimension. It also defines special types of matrices such as zero matrices, triangular matrices, and identity matrices. The document then explains how to perform basic matrix operations like addition, scalar multiplication, and multiplication. It establishes properties for these operations and provides examples to illustrate them.

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F
ก
““““ F F”””” F 7
F
F F F F . . 2537
www.thai-mathpaper.net

3. F F ก F F ˈ F F 7 15 F
F F F F F F F ˈ ก
ก ก F F F ก ก ก F F F ก ก F
ก F F ก F
F ก 1 ˈ ก F F ก ก F F ก ก
ก F ก F ก ก F 2 × 2
2 ˈ ก F F ก F F ˈ ก กF ˆ F F
ก F ก F F ก F F ก ก
F F F F F ก F ก ก F n × n n ≥ 2
3 ˈ F F F F ก F ก ก F F ก F
F F ก กF ˆ F F F F ก กF ก F
ก F ก Fก ก F ก
F ˈ F F F F ˈ F F Fก ก
F ก ก F ก F F F F F F ก F F F
F F F F F F
F
8 ก . . 2549

5. 1 ก F 1 13
1.1 ก F 1
1.2 ก F 2
1.3 ก F ก F 3
1.4 ก ก F F ก F 6
1.5 ก ก ก F 8
1.6 F ก ก F 2 × 2 10
2 F F 15 25
2.1 F F 15
2.2 F F ก F กก F 3 × 3 17
2.3 F F 20
2.4 F ก ก F กก F 3 × 3 23
3 ก ก F ก F F F 27 38
3.1 ก ก (Row operation) 27
3.2 ก กF ก F 29
3.3 ก F ก ก F Fก ก 36
ก 39

7. 1
ก F
1.1 ก F
F ก F
F F F F F F ก F F F ก F ก F
ก F ก F ก (element) ก F ก (column)
F ก F (row)
ก F ก F ก F F ก F F ก ก F
ก ก F ก ก F ก 2 ก F (index) F F ก
ก F F F ก ก ก F
ก กก F (dimension) ก F ( ) ก F
F ก ก ก F ก ก ก
ก
ก F ก 1 F ก กก F 1 ก ก F ก F (row
matrix) ก ก F ก 1 ก F กก F 1 ก F
ก F ก (column matrix)
กF F ก ก ก F F F F
ก F ก ก FกF F
1.1
ก F (Matrices) ก ก F F กF [ ]
() F ก F F F ก F
1.2
ก F F ก F A F ก ก F B F A = B ก F ก F F ก
ก F F ก F F ก

8. 2 ก F
F 1.1 ก F A =
1 2
3 0
 
 
 
, B =
1
0
 
 
 
, C =
11 12 13
21 22 23
31 32 33
a a a
a a a
a a a
 
 
 
 
 
F
1) ก F 11 F 21 ก F A
2) ก F B ˈ ก F ( ก F ก ก F ) F ก B
3) ก F C
1) ก F A ก F 11 a11 1 ก F 21 a21
3
2) ก F B F ˈ ก F ก F ก 2 × 1
3) ก F C F ก 3 × 3
1.2 ก F
F ก F ก F F F
F
1.3
1) ก F F (zero matrix) ก F ก ก ˈ F F 0
2) ก F (triangular matrix) ก F ก F F F
ก ˈ F F ก ˈ 2 ก F ก ก F F
3) ก F (square matrix) ก F F ก ก
4) ก F ก ก F (identity matrix) ก F n × n ก F
ก ˈ ก F ˈ F F In
5) ก F ก F (Transposition of matrix) ก F A F AT
ก F F กก ก ก ก F

9. F F 3
ʿก 1.2
1. ก F 1.1 F ก F F F F ก
2. ก F ก F F กก F 2 × 2 ก F ก ก F 3 × 3
1 F
3. ก F A =
1 0 -1
2 1 0
3 2 -1
 
 
 
 
 
ก F F (AT
) A ( F )
1.3 ก F ก F
F ก ก ก
ก ก ก F ก ก ก ก F ก ก ก ก Fก ก
ก ก ก Fก ก F ก ก ก F ก ก F
ก F ก F ก F ก ก F FกF F ก F F F F
F F ก ก ก F กF
1.3.1 ก ก ก ก F
F ก
1. ก 1.3 F ก F 2 ก ก F F F ก
2. ก ก ก F F C = A + B ก F F ก F C ก F ก
F ก ก ก F A ก F B F F ก ก ก
1.4
ก F A = [aij], B = [bij], C = [cij] ˈ ก F F ก m × n F
1) A + B = [aij + bih]
2) A B = [aij bih]

10. 4 ก F
F 1.2 ก F A =
1 2
0 -1
 
 
 
, B =
2 0
1 -3
 
 
 
, C =
1
1
 
 
 
A + B, A + C, B + C
ก 1.3 F F A + B =
1 2
0 -1
 
 
 
+
2 0
1 -3
 
 
 
=
1 + 2 2 + 0
0 + 1 (-1) + (-3)
 
 
 
=
3 2
1 -4
 
 
 
A + C =
1 2
0 -1
 
 
 
+
1
1
 
 
 
F ก F A ก F C F F ก
F F A + C F B + C ก F F F F ก
F 1.3 ก F F 1.2 F A B
ก 1.2 F F A B =
1 2
0 -1
 
 
 
2 0
1 -3
 
 
 
=
1 - 2 2 - 0
0 - 1 (-1) - (-3)
 
 
 
=
-1 2
-1 2
 
 
 
1.3.2 ก ก F F ก F
ก 1.5 F F F F ก ก F F ก F ก ก F ก F
ก ก ก ก F ก F ก
F 1.4 ก F A =
1 2
3 4
 
 
 
k = 1 F F kA
ก kA = k
1 2
3 4
 
 
 
=
k 2k
3k 4k
 
 
 
F k = 1 F F kA =
-1 -2
-3 -4
 
 
 
1.5
ก F c ˈ F ˈ 0 A = [aij]m × n F F cA = [caij]m × n

11. F F 5
1.3.3 ก F
F F A = [aij] , B = [bij] , C = [cij]
1) A + B = [aij] + [bij]
= [aij + bij]
= [bij + aij]
= [bij] + [aij]
= B + A
2) c(A + B) = c[aij + bij]
= [caij + cbij]
= [caij] + [cbij]
= cA + cB
3) A + (B + C) = [aij] + [bij + cij]
= [aij + bij + cij]
= [aij + bij] + [cij]
= (A + B) + C
4) A + 0 = [aij] + [0ij]
= [aij + 0ij]
= [0ij + aij]
= [0ij] + [aij]
= 0 + A
F ก F 0 ˈ ก ก Fก ก F F [aij + 0ij]
[0ij + aij] F F ก [aij] = A
5) A + ( A) = (1)A + ( 1)A = (1 1)A = 0A = 0
1.1
ก F A, B, C ˈ ก F m × n c ˈ F F
1) A + B = B + A ก F ก ก
2) c(A + B) = cA + cB ก F ก ก F ก F
3) A + (B + C) = (A + B) + C ก F ก ก F ก ก
4) A + 0 = 0 + A = A ก F ก ก ก Fก ก
5) A + ( A) = 0 ก F ก F ก ก

12. 6 ก F
ʿก 1.3
1. ก ก F A, B F 1.2 F 2A B, B 2A, 2A 2B B A
1.4 ก ก F F ก F
ก ก F F F F F ก ก ก
ก F F ก F ก F กก F Fก
ก ก F F กF F F F ก ก F F ก F F
F ก
ก F ก F ก F ก ก F F ก ก F
ก F F ก ก F ก F ก ก
ก F
F 1.5 ก F A =
1 2
-1 0
 
 
 
, B =
3 -2
0 -4
 
 
 
AB BA
F ก F A 2 × 2 ก F B 2 × 2 F ก
ก F A F ก ก F B ก F
AB =
1 2
-1 0
 
 
 
3 -2
0 -4
 
 
 
=
(1)(3) + (2)(0) (1)(-2) + (2)(-4)
(-1)(3) + (0)(0) (-1)(-2) + (0)(-4)
 
 
 
=
3 -10
-3 2
 
 
 
1.5
ก F A = [aij]m×p B = [bij]p×n F F ก F A ก F B F AB
ก AB = C = [cij]m×n cij = ai1b1j + ai2b2j + ai3b3j + + aipbpj

13. F F 7
BA =
3 -2
0 -4
 
 
 
1 2
-1 0
 
 
 
=
(3)(1) + (-2)(-1) (3)(2) + (-2)(0)
(0)(1) + (-4)(-1) (0)(2) + (-4)(0)
 
 
 
=
5 6
4 0
 
 
 
ก F F F AB ≠ BA F AB ≠ BA F F ก
ก F AB = BA F
ก ก ก F F ก F F F ˈ ก ก ก ก F
F F ก F (2)
F A = [aij], B = [bij], C = [cij] i = 1, 2, 3, , m j = 1, 2, 3, , n
A(B + C) = [aij] ([bij] + [cij])
= [aij] ([bij + cij])
= [ai1(b1j + c1j) + ai2(b2j + c2j) + ai3(b3j + c3j) + + aip(bpj + cpj)]
= [ai1b1j + ai1c1j + ai2b2j + ai2c2j + ai3b3j + ai3c3j + + aipbpj + aipcpj]
= [(ai1b1j + ai2b2j + ai3b3j + aipbpj) + (ai1c1j + ai2c2j + ai3c3j + + aipcpj)]
= [(ai1b1j + ai2b2j + ai3b3j + aipbpj)] + [(ai1c1j + ai2c2j + ai3c3j + + aipcpj)]
= AB + AC
ʿก 1.4
1. F 1.2 (1) ก (3) ˈ
1.2 ก F A, B, C ˈ ก F m × n F F
1) (AB)C = A(BC)
2) A(B + C) = AB + AC
3) (B + C)A = BA + CA

14. 8 ก F
1.5 ก ก ก F
F 1) cAT
= c[aij]T
= c[aji]
= [caji]
= [caij]T
= (cA)T
2) (AT
)T
= { }
TT
aij  
=
T
aji  
= [aij]
3) (A + B)T
= ([aij] + [bij])T
= ([aij + bij])T
= [aji + bji]
= [aji] + [bji]
= [aij]T
+ [bij]T
F ก : F B F B F (3) ก ˈ
4) (AB)T
= ([aij] [bij])T
= ([ai1b1j + ai2b2j + ai3b3j + + a1pbpj])T
= ([b1jai1 + b2jai2 + b3jai3 + + bpja1p])T
= ([bj1a1i + bj2a2i + bj3a3i + + bjpap1])
= BT
AT
5) (c + d)A = (c + d)[aij]
= c[aij] + d[aij]
= [caij] + [daij]
1.3 ก F A = [aij], B = [bij], C = [cij] ˈ ก F m × n c , d ˈ
1) cAT
= (cA)T
2) (AT
)T
= A
3) (A + B)T
= AT
+ BT
4) (AB)T
= BT
AT
5) (c + d)A = cA + dA
6) c(dA) = (cd)A = d(cA)

15. F F 9
= cA + dA F ก : F d F d F (5) ก ˈ
6) c(dA) = c(d[aij]) =
F 1.6 A2
= AA
A3
= A2
A
A4
= A2
A2
= A3
A
F 1) ก (A + B)2
= (A + B)(A + B)
= ([aij + bij])( [aij + bij])
= [(ai1 + bi1)(a1j + b1j) + (ai2 + bi2)(a2j + b2j) + + (aip + bip)(apj + bpj)]
= [(ai1a1j + bi1a1j + ai1b1j + bi1b1j) + (ai2a2j + bi2a2j + ai2b2j + bi2b2j) + +
(aipapj + bipapj + aipbpj + bipbpj)]
= [ai1a1j + ai2a2j + + aipapj] + [bi1a1j + bi2a2j + + bipapj] +
[ai1b1j + ai2b2j + + aipbpj] + [bi1b1j + bi2b2j + + bipbpj]
= A2
+ BA + AB + B2
2) F ก (1) F ก B F B
3) ก (A + B)(A B) = ([aij + bij])([aij bij])
= [(ai1 + bi1)(a1j b1j) + (ai2 + bi2)(a2j b2j) + + (aip + bip)(apj bpj)]
= [(ai1a1j + bi1a1j ai1b1j + bi1b1j) + (ai2a2j + bi2a2j ai2b2j + bi2b2j) +
(aipapj + bipapj aipbpj + bipbpj)]
= [ai1a1j + ai2a2j + aipapj] + [bi1a1j + bi2a2j + + bipapj]
[ai1b1j + ai2b2j + + aipbpj] + [bi1b1j + bi2b2j + + bipbpj]
= A2
+ BA AB + B2
1.4 ก F A = [aij], B = [bij], C = [cij] F F
1) (A + B)2
= A2
+ AB + BA + B2
2) (A B)2
= A2
(AB + BA) + B2
3) (A + B)(A B) = A2
+ BA AB + B2
1.6 F A ˈ ก F n ˈ ก F F An
= AAA A
n ก F
(cd)[aij] = (cd)A
d(c[aij]) = d(cA)

16. 10 ก F
ก F 1.4 (1) ก (2) ก Fก ก F F F ก
ก F ก F F ก F F F ก
ก F F ก ก F
ก (A + B)2
= (A + B)(A + B)
= AA + BA + AB + BB = A2
+ BA + AB + B2
(A B)2
= (A B)(A B)
= AA BA + AB + BB = A2
BA + AB + B2
1.6 F ก ก F 2 ×××× 2
กF F ก ก F n × n F
F ก ก F 2 × 2 ˈ ก F ก ก F n × n
กF F
1.7
F A ˈ ก F 2 × 2 A =
a b
c d
 
 
 
ก F F A 1
ˈ F ก
ก F A ก F ad bc ≠ 0 F ad bc ≠ 0 F F ก ก F A F ก ก
A 1
=
d -b
1
ad - bc -c a
 
 
 

17. F F 11
ก F F F F F F ก F ก F F F F F F
F ก F F ก F F ˈ ก F (square matrix) F ก ก F
F ก F F ก F F ก (non singular matrix) ก F F
F ก F ก F ก F ก (singular matrix)
F 1.7 ก F A =
1 2
3 4
 
 
 
F ก F A F ก F F F F
F F ก
ก ad bc = (1)(4) (2)(3) = 2 ≠ 0 F A F ก F
ก ก A 1
=
d -b
1
ad - bc -c a
 
 
 
= 1
-2
4 -2
-3 1
 
 
 
= 3 1
2 2
-2 1
-
 
 
 
F ก ก F 2 × 2 F ก F
F F ก F 3) F F F F F ˈ ʿก
F A =
11 12
21 22
a a
a a
 
 
 
, B =
11 12
21 22
b b
b b
 
 
 
ก AB =
11 12
21 22
a a
a a
 
 
 
11 12
21 22
b b
b b
 
 
 
1.5
ก F A ˈ ก F ก 2 × 2 A 1
ˈ F ก A F
1) AA 1
= A 1
A = I2
2) (A 1
) 1
= A
3) (AB) 1
= B 1
A 1
4) (AT
) 1
= (A 1
)T
5) (Ak
) 1
= (A 1
)k
k ˈ ก

18. 12 ก F
=
11 11 12 21 11 12 12 22
21 11 22 21 21 12 22 22
a b + a b a b + a b
a b + a b a b + a b
 
 
 
ก ก F ก F a11b11 + a12b21 = m
a11b12 + a12b22 = n, a21b11 + a22b21 = p a21b12 + a22b22 = q
F F AB =
m n
p q
 
 
 
F F (AB) 1
=
q -n
1
mq - pn -p m
 
 
 
-----(1.6.1)
B 1
A 1
ก B 1
=
11 22 21 12
22 121
b b - b b
21 11
b -b
-b b
 
 
 
A 1
=
11 22 21 12
22 121
a a - a a
21 11
a -a
-a a
 
 
 
F F
B 1
A 1
=
11 22 21 12
22 121
b b - b b
21 11
b -b
-b b
  
  
   11 22 21 12
22 121
a a - a a
21 11
a -a
-a a
  
  
  
= ( )( ){ }11 22 21 12 11 22 21 12
1 1
b b - b b a a - a a
22 12 22 12
21 11 21 11
b -b a -a
-b b -a a
   
   
   
=
( )( )11 22 21 12 11 22 21 12
1
b b - b b a a - a a
22 12 22 12
21 11 21 11
b -b a -a
-b b -a a
   
   
   
=
11 22 11 22 21 12 11 22 11 22 21 12 21 12 21 12
1
b b a a - b b a a - b b a a + b b a a
q -n
-p m
 
 
 
=
q -n
1
mq - pn -p m
 
 
 
-----(1.6.2)
F F ก (1.6.1) = ก (1.6.2) F ก F ก F ˈ
F F F F b11b22a11a22 b21b12a11a22 b11b22a21a12 + b21b12a21a12 F F ก
mq pn
22 12 22 12
21 11 21 11
b -b a -a
-b b -a a
   
   
   
=
q -n
-p m
 
 
 

19. F F 13
F ก F A B F AB ≠ BA F ก A, B ˈ ก F
2 × 2 A F ก F F F F F B = A 1
F AB = BA F
F F A, B ˈ ก F 2 × 2 A F ก
F B = A 1
A F F F ก F F AB = AA 1
= I
A F F ก F F BA = A 1
A = I
AB = BA
1.5 1.6 ˈ ก F กก F 2 × 2
F F F F F F ก ก ก F (Linear Algebra)
ก
ʿก 1.6
1. ก F ก F ก (singular matrix) 3 F
2. ก ก F A, B F 1.2 F 1.7 ก F ก F A, B F ก
F F ก A, B F ก 1.5
3. F ก F F (zero matrix) 2 × 2 ˈ ก F ก
4. ก F I2 ˈ ก F ก ก F 2 × 2 ก F ก ก F I2
1.6
ก F A, B ˈ ก F 2 × 2 A F ก F F F B = A 1
F AB = BA

21. 2
F F
2.1 F F
ก ก F F F ก กF ˆ F F ˈ F ก F
ก F ก F ก F F (determinant)
ก F F ก F 1 × 1, 2 × 2 3 × 3
ก F A1 = [a11] F F det(A1) = a11 ก F F F F ก F
ก F F ก ก
ก F A2 =
11 12
21 22
a a
a a
 
 
 
F F det(A2) = a11a22 a21a12
ก F A3 =
11 12 13
21 22 23
31 32 33
a a a
a a a
a a a
 
 
 
 
 
F F
det(A3) = (a11a22a33 + a12a23a31 + a13a32a21) (a13a31a22 + a32a23a11 + a33a21a12)
ก F ก F ก F F ก ˈ ก F กก ก F
F F ก F ก F ก ก ก F
F F ก F F ก F F FกF ก ก F F
F F F กF
2.1
F F ˆ กF ก ก F n × n
F F ก F A F det(A) F F F |A| ก F

22. 16 ก F
F 2.1 ก F A =
1 -3 2
5 4 0
1 1 2
 
 
 
 
 
F det(A), M23(A), C23(A)
1) ก ก F A ก ˈ ก F 3 × 3 F ก
det(A3) = (a11a22a33 + a12a23a31 + a13a32a21) (a13a31a22 + a32a23a11 + a33a21a12) -----(2.1.1)
F ก F ก F A ก (2.1.1) F F
det(A)= [(1)(4)(2) + ( 3)(0)(1) + (2)(1)(5) (2)(1)(4) + (1)(0)(1) + (2)(5)( 3)]
= (8 + 0 + 10) (8 + 0 30)
= 18 ( 22)
= 40
2) ก M23(A) F F ก F F กก 2 ก 3
ก F A ก
M23(A) =
1 -3
1 1
ก M23(A) ˈ ก F 2 × 2 F ก det(A2) = a11a22 a21a12
F F det(A) = M23(A) = (1)(1) (1)( 3) = 1 + 3 = 4
3) ก C23(A) = ( 1)2 + 3
M23(A)
F M23(A) ก F 2) F F C23(A) = ( 1)(4) = 4
2.2 ก F A ˈ ก F n × n n ≥ 2
1) F (minor) ก F (i, j) ก F A F F F กก
i ก j ก F A ก F Mij(A)
2) ก F (cofactor) ก F (i, j) ก F A F F ก ( 1)i + j
Mij(A)
F Cij(A)
3) ก F ก F F F ก F A F ก F cof(A)
F F F C(A) ก F

23. F F 17
ʿก 2.1
1. ก F A ˈ ก F 3 × 3 F F F ก F
ก F F F ก ก F ก ก F
2. F det(I3) = 1
3. F det(03) = 0
4. ก F B =
1 0 0
-2 3 1
3 -1 5
 
 
 
 
 
ก F det(B), M31(B) C31(B)
5. ก F C =
11 12
21 22
c c
c c
 
 
 
F C 1
=
22 121
det(C)
21 11
c -c
-c c
 
 
 
2.2 F F ก F กก F 3 ×××× 3
F F F ก F F F ก F 1 × 1, 2 × 2 3 × 3
F Fก F F ก F F F ก F Fก F F กก F
F ก ก F ˈ F F ก F F กก i ก j ก F ก
ก F F F ก ( 1)i + j
Mij(A) F F ก F F ก F
F F ก F กก F 3 × 3
ก 2.1 ก ก 1) F ก ก ก F ก ก 2) F ก
ก ก F ก ก ก F ก 1) 2) ก F F F F ก
F ก F F F F F ก F F F F
ก ก ก F (Linear Algebra) ก
2.1 ก F A ˈ ก F n × n n ≥ 3 F F
1) det(A) = a11C11(A) + a12C12(A) + a13C13(A) + + a1nC1n(A)
2) det(A) = a11C11(A) + a21C21(A) + a31C31(A) + + an1Cn1(A)

24. 18 ก F
F ก
1. 2.1 F ก ก F 1 ก 1 ก
ก i ≤ n ก j ≤ n ก F
2. ก Fก ก ก F i ≤ n ก j ≤ n
ก ˈ F กก F
F 2.2 ก F A =
-1 2 3 1
3 1 0 1
0 1 0 0
1 2 -1 5
 
 
 
 
 
 
F det(A)
ก ก F A ก F F ก ก ก F 3 ก ก F
F ก
ก ก ก F 3
ก det(A) = a31C31(A) + a32C32(A) + a33C33(A) + a34C34(A) -----(2.2.1)
F a31 = a33 = a34 = 0 ก (2.2.1) F ˈ det(A) = a32C32(A) -----(2.2.2)
C32(A) = ( 1)3 + 2
M32(A)
= M32(A) =
-1 3 1
3 0 1
1 -1 5
-----(2.2.3)
Fก ก ก F 2 ก ก (2.2.3)
ก
-1 3 1
3 0 1
1 -1 5
= a21C21(A) + a23C23(A) ( ก a22 = 0 F ก a22C22(A))
= 3( 1)2 + 1
M21(A) + 1( 1)2 + 3
M23(A)
= 3( 1)(16) + 1( 1)( 2) = 48 + 2 = 46
F F det(A) = a32C32(A) = (1)( 1)( 46) = 46
F 2.3 ก F F 2.2 F det(A) Fก ก ก F ก 3
ก ก ก F ก 3
ก det(A) = a13C13(A) + a23C23(A) + a33C33(A) + a43C43(A) -----(2.2.4)
F a23 = a33 = 0 ก (2.2.4) F ˈ det(A) = a13C13(A) + a43C43(A) ----(2.2.5)
C13(A) = ( 1)1 +3
M13(A)

25. F F 19
= M13(A) =
3 1 1
0 1 0
1 2 5
-----(2.2.6)
Fก ก ก F 2 ก ก (2.2.6)
ก
3 1 1
0 1 0
1 2 5
= a22C22(A)
= (1)( 1)2 + 2
3 1
1 5
= 14
C43(A) = ( 1)4 + 3
M43(A)
= M43(A)
=
-1 2 1
3 1 1
0 1 0
-----(2.2.7)
Fก ก ก F 3 ก ก (2.2.7) F
ก
-1 2 1
3 1 1
0 1 0
= a32C32(A) =
-1 1
3 1
= ( 1 3) = 4
F F det(A) = a13C13(A) + a43C43(A) = (3)(14) + ( 1)( 4) = 42 + 4 = 46
ʿก 2.2
1. ก F f ˈ ˆ กF ก ก F 3 × 3 F f
F ˈ ˆ กF F
2. ก F A =
1 2 1 0
0 1 1 3
0 0 1 2
x 1 1 1
 
 
 
 
 
 
x ˈ F F det(A) = 6 F
C41(A) + C42(A) + C43(A) + C44(A)

26. 20 ก F
2.3 F F
F F ก F (3), (4), (9), (11) F F F F F F
F F F ก ก ก F ก
กF ก F A =
11 12 13 1n
21 22 23 2n
31 32 33 3n
n1 n2 n3 nn
a a a a
a a a a
a a a a
a a a a
 
 
 
 
 
 
  
⋯
⋯
⋯
⋮ ⋮ ⋮ ⋮ ⋮
⋯
3) Fก ก ก F 1 F F
det(A) = a11C11(A) + a12C12(A) + a13C13(A) + + a1nCin(A) -----(2.3.1)
2.2
ก F A, B ˈ ก F n × n
1) det(AB) = det(A)⋅det(B)
2) det(A 1
) = 1
det(A) det(A) ≠ 0
3) det(AT
) = det(A)
4) det(In) = 1
5) det(Ak
) = [det(A)]k
k ˈ F F ก 0
6) det(kA) = kn
⋅det(A) k ˈ
7) F A ก ก ก ˈ F F det(A) = 0
8) F A ก ก ก F det(A) = 0
9) F B ˈ ก F ก กก F k ก ก ก
ก ก F A F det(B) = k⋅det(A)
10) F B ˈ ก F ก กก ก ก ก F A F
det(B) = det(A)
11) F B ˈ ก F ก กก F k ก ก ก
ก F กก ก ก ก F A F det(B) = det(A)

27. F F 21
F AT
F F AT
=
11 21 31 n1
12 22 32 n2
13 23 33 n3
1n 2n 3n nn
a a a a
a a a a
a a a a
a a a a
 
 
 
 
 
 
  
⋯
⋯
⋯
⋮ ⋮ ⋮ ⋮ ⋮
⋯
Fก ก ก F ก 1 F
det(AT
) = a11C11(A) + a12C12(A) + a13C13(A) + + a1nCin(A) -----(2.3.2)
F ก (2.3.1) = ก (2.3.2) F F det(A) = det(AT
)
4) ก F In =
1 0 0 0
0 1 0 0
0 0 0
0
0 0 0 1
 
 
 
 
 
 
  
⋯
⋯
⋯ ⋯
⋮ ⋮ ⋮ ⋮
⋯
Fก ก ก F 1 F F det(In) = 1
9) F k ˈ k 1 F F
A =
11 12 1n
21 22 2n
31 32 3n
n1 n2 nn
ka ka ka
a a a
a a a
a a a
 
 
 
 
 
 
  
⋯ ⋯
⋯ ⋯
⋯ ⋯
⋮ ⋮ ⋮ ⋮ ⋮
⋯ ⋯
Fก ก ก F 1 F F
det(A) = ka11C11(A) + ka12C12(A) + ka13C13(A) + + ka1nCin(A)
= k(a11C11(A) + a12C12(A) + a13C13(A) + + a1nCin(A))
= k⋅det(A)

28. 22 ก F
11) F k ˈ k ก 3 F กก 1 F
B =
31 11 32 12 3n 1n
21 22 2n
31 32 3n
n1 n2 nn
ka + a ka + a ka + a
a a a
a a a
a a a
 
 
 
 
 
 
  
⋯ ⋯
⋯ ⋯
⋯ ⋯
⋮ ⋮ ⋮ ⋮ ⋮
⋯ ⋯
Fก ก ก F 1 F
= ka31C11(A) + a11C11(A) + ka32C12(A) + a12C12(A) + ka33C13(A) + a13C13(A) +
+ ka3nC1n(A) + a1nC1n(A)
= (ka31C11(A) + ka32C12(A) + ka33C13(A) + + ka3nC1n(A)) + (a11C11(A) + a12C12(A)
+ a13C13(A) + + a1nC1n(A))
= k(a31C11(A) + a32C12(A) + a33C13(A) + + a3nC1n(A)) + det(A)
F k(a31C11(A) + a32C12(A) + a33C13(A) + + a3nC1n(A)) = 0 det(B) = det(A)
ʿก 2.3
1. F 2.2 F ˈ
2. ก F A, B ˈ ก F 3 × 3 det(A), det(B) ≠ 0 F F F F
det(AB) 1
= 1
det(A) det(B)⋅
3. ก F A ˈ ก F 3 × 3 det(A) ≠ 0 F F F F F
det (kA 1
) =
3
k
det(A) k ≠ 0

29. F F 23
2.4 F ก ก F กก F 3 ×××× 3
F 1.6 F ก F ก ก F 2 × 2 F F
ก F ก ก F กก F 3 × 3 FกF F F
F กF
F 2.4 ก F A =
3 -1 2
0 1 5
1 2 1
 
 
 
 
 
adj(A)
ก adj(A) = [cof(A)]T
F [cof(A)]T
=
11 21 31
12 22 32
13 23 33
C (A) C (A) C (A)
C (A) C (A) C (A)
C (A) C (A) C (A)
 
 
 
 
 
C11(A) = M11(A) =
1 5
2 1
= (1)(1) (2)(5) = 1 10 = 9
C12(A) = M12(A) =
0 5
1 1
= [(0)(1) (1)(5)] = 5
C13(A) = M13(A) =
0 1
1 2
= (0)(2) (1)(1) = 1
C21(A) = M21(A) =
-1 2
2 1
= [( 1)(1) (2)(2)] = 5
C22(A) = M22(A) =
3 2
1 1
= (3)(1) (1)(2) = 1
C23(A) = M23(A) =
3 -1
1 2
= [(3)(2) (1)( 1)] = 7
C31(A) = M31(A) =
-1 2
1 5
= ( 1)(5) (1)(2) = 7
C32(A) = M32(A) =
3 2
0 5
= [(3)(5) (0)(2)] = 15
2.3
ก F ก (adjoint matrix) ก F A n × n n ≥ 2 ก F ก F
ก F ก F A F ก F [cof(A)]T
adj(A)

30. 24 ก F
C33(A) = M33(A) =
3 -1
0 1
= (3)(1) (0)( 1) = 3
adj(A) = [cof(A)]T
=
-9 5 -7
5 1 -15
-1 -7 3
 
 
 
 
 
F ก
ก ก n = 2 ˈ ก F ก F ก F 1.6 F ก
F กก F
F 2.5 ก ก F A F 2.4 A 1
ก ก A 1
=
adj (A)
det (A) ก F 2.4 F adj(A) =
-9 5 -7
5 1 -15
-1 -7 3
 
 
 
 
 
det(A) ก F F ก
det
3 -1 2
0 1 5
1 2 1
  
  
  
  
  
Fก ก ก F 2 F F ก 34
A 1
=
-9 5 -7
1 5 1 -15-34
-1 -7 3
 
 
 
 
 
=
9 5 7
34 34 34
5 151
34 34 34
7 31
34 34 34
-
- -
-
 
 
 
 
  
F F F AA 1
= A 1
A = I F F ก ก F
ก ก F กก F 3 × 3 ก F ก 1.5 F F F F F ก
F Fก F ก
ก F ก ก F ก ก F F ก F ก F F
ก ก F F
2.4
ก F A ˈ ก F n × n n ≥ 2 F det(A) ≠ 0 F F F
A 1
= adj (A)
det (A)

31. F F 25
ʿก 2.4
1. ก F A =
3 0 1 3
6 2 3 -2
1 0 -1 0
1 0 1 0
 
 
 
 
 
 
1) ก F ก (cof(A))
2) F ก A (A 1
)
2. ก F 2.4 F 2.5 F AA 1
= A 1
A = I3
3. ก F A ˈ ก F 3 × 3 det(A) ≠ 0 F det(adj(A)) = [det(A)]2
ก F n × n F

33. 3
ก ก F ก F F F
ก ก F ก F F F F ก F ก F ก
ก F ก กF ก F ก F ก ก F n × n
3.1 ก ก (Row operation)
กF ก ก ก ก F F ก F F F ก กF ˆ F F
F F ก F ก ก F ก ก F F F F ก
F F
ก ก (row equivalence) ก (column equivalence)
F ก 3.1 3.2 F ก กก ก F ก ก ก (column
operation)
3.1
ก ก ก F A F กF ก ก F F F
1) F i ก j F ก F Ri ↔ Rj
2) i F k ≠ 0 F ก F kRi
3) i F k ≠ 0 F กก j i ≠ j F ก F
Rj + kRi
3.2
ก F A, B ˈ ก F m × n ก F F A ก B ก F B F กก
ก ก F A F ก F
A row B

34. 28 ก F
F 3.1 ก F A =
1 1 2
1 2 2
2 1 1
 
 
 
 
 
F A ∼ I3
ก
1 1 2
1 2 2
2 1 1
 
 
 
 
 
2 2 1
3 3 1
R R - R
R R - 2R
→
→
→
1 1 2
0 1 0
0 -1 -3
 
 
 
 
 
3 3 2R R + R→
→
1 1 2
0 1 0
0 0 -3
 
 
 
 
 
3 3 2R R + R→
→
1 1 2
0 1 0
0 0 -3
 
 
 
 
 
( )1
3 33R - R→
→
1 1 2
0 1 0
0 0 1
 
 
 
 
 
1 1 3R R - 2R→
→
1 1 0
0 1 0
0 0 1
 
 
 
 
 
1 1 2R R - R→
→
1 0 0
0 1 0
0 0 1
 
 
 
 
 
= I3 A ∼ I3 F ก
ʿก 3.1
1. F ก F B ʿก 2.1 F 4. ก I3 F
2. F ก F A F 2.4 ก I3 F
3. F ก F A ʿก 2.4 F 1. ก I4 F

35. F F 29
(3.2.2) ˈ ก F 5 x, y, z, s, t
(3.2.1) ˈ ก F 3 x, y, z
3.2 ก กF ก F
ก F F ก ก ก ก กF ก F 2 F
Fก กF ก F F ก F F F F ก กF
ก F 2 ก ˈ ก ก กF ก F
3.2.1 ก F
F 3.2 F ก F
1) ก F
2x + 3y + z = 1
x + y z = 0
x 2y + z = 2
2) ก F F
x + y + z + s + t = 5
x y + z + s t = 1
x y z s + t = 1
x + y + z s t = 1
x y z s t = 5
3.3
ก F (Multivariable System of Linear Equations) ก F
a11x1 + a12x2 + a13x3 + + a1nxn = b1
a21x1 + a22x2 + a23x3 + + a2nxn = b1
a31x1 + a32x2 + a33x3 + + a3nxn = b1
an1x1 + an2x2 + an3x3 + + annxn = bn
a11, a12, a13, , ann ก F
b1, b2, b3, , bn ˈ ก F F

36. 30 ก F
3.2.2 ก F
ก กF ก F 2 ก F ก ˈ 3 ก F กF
1)
2)
3) F
ก F ก ก F
F ก F ก n F
ก ก F F ก ก F ก
n F ก ก ก F F F
ก ก F F ก n F F ก ก ก
F F F ก F F F ก
3.2.3 ก ก F F ก F
F F ก ก F ก 3.3 F F ก
F ก F ก F ก ก F F
3.4
ก F a11x1 + a12x2 + a13x3 + + a1nxn = b1
a21x1 + a22x2 + a23x3 + + a2nxn = b1
a31x1 + a32x2 + a33x3 + + a1nxn = b1
an1x1 + an2x2 + an3x3 + + annxn = bn
F F AX = B F A ก F ก , X
ก F F F ก B ก F F ก
A =
11 12 13 1n
21 22 23 2n
31 32 33 3n
n1 n2 n3 nn
a a a a
a a a a
a a a a
a a a a
 
 
 
 
 
 
  
⋯
⋯
⋯
⋮ ⋮ ⋮ ⋮ ⋮
⋯
, X =
1
2
3
n
x
x
x
x
 
 
 
 
 
 
  
⋮
B =
1
2
3
n
b
b
b
b
 
 
 
 
 
 
  
⋮

37. F F 31
F 3.3 ก F F ก F
3m + 2n p + q = 4
m + 2n 2p + 3q = 6
n 3p q = 1
2m + 5n p + 3q = 3
ก ก Fก F F ˈ 4 ก 4 F F
A =
3 2 -1 1
1 2 -2 3
0 1 -3 -1
2 5 -1 3
 
 
 
 
 
 
, X =
m
n
p
q
 
 
 
 
 
 
, B =
4
6
1
3
 
 
 
 
 
 
F ก ก F 3.3 F ก 3 F ก F m F m 0 ˈ
ก F ก ก F 0 ˈ
ʿก 3.2 ก
ก F F ก F 3.2
3.2.4 ก กF ก F Fก F
ก กF ก F F F F 2
F ก Fก F
F F F F F F F F
F ก ก ก F ก
3.1
ก F AX = B ˈ ก F n ก n det(A) ≠ 0 F
ก
x1 = 1det(A )
det(A) , x2 = 2det(A )
det(A) , x3 = 3det(A )
det(A) , , xj = jdet(A )
det(A) , xn = ndet(A )
det(A)
Aj ก F ก กก ก j ก F A F ก F B

38. 32 ก F
F 3.4 กF ก 3x + 2y = 1
x + y = 0
ก ก ก F F ก F F
3 2
1 1
 
 
 
x
y
 
 
 
=
1
0
 
 
 
-----(3.2.3)
F A =
3 2
1 1
 
 
 
F F det(A) = (3)(1) (1)(2) = 1
A1 =
1 2
0 1
 
 
 
F det(A1) = (1)(1) (0)(2) = 1
A2 =
3 1
1 0
 
 
 
F det(A2) = (3)(0) (1)(1) = 1
x1 = 1det(A )
det(A) = 1
1 = 1, x2 = 2det(A )
det(A) = -1
1 = 1
F F ก (x, y) = (1, 1)
F 3.5 กF ก x + 2y + z = 5
2x 5y 5z = 0
3x + 2y 3z = 1
ก ก ก F F ก F F
1 2 1
2 -5 -5
3 2 -3
 
 
 
 
 
x
y
z
 
 
 
 
 
=
5
0
-1
 
 
 
 
 
-----(3.2.4)
F A =
1 2 1
2 -5 -5
3 2 -3
 
 
 
 
 
F det(A) =
1 2 1 1 2
2 -5 -5 2 -5
3 2 -3 3 2
= (1)( 5)( 3) + (2)( 5)(3) + (1)(2)(2) (3)( 5)(1) (2)( 5)(1) ( 3)(2)(2)
= 15 30 + 4 + 15 + 10 + 12 = 26
A1 =
5 2 1
0 -5 -5
-1 2 -3
 
 
 
 
 
F det(A1) =
5 2 1 5 2
0 -5 -5 0 -5
-1 2 -3 -1 2
= (5)( 5)( 3) + (2)( 5)( 1) + (1)(0)(2) ( 1)( 5)(1) (2)( 5)(5) ( 3)(0)(2)
= 75 + 10 5 + 50 = 130

39. F F 33
A2 =
1 5 1
2 0 -5
3 -1 -3
 
 
 
 
 
F det(A2) =
1 5 1 1 5
2 0 -5 2 0
3 -1 -3 3 -1
= (1)(0)( 3) + (5)( 5)(3) + (1)(2)( 1) (3)(0)(1) ( 1)( 5)(1) ( 3)(2)(5)
= 75 2 5 + 30 = 52
A3 =
1 2 5
2 -5 0
3 2 -1
 
 
 
 
 
F det(A3) =
1 2 5 1 2
2 -5 0 2 -5
3 2 -1 3 2
= (1)( 5)( 1) + (2)(0)(3) + (5)(2)(2) (3)( 5)(5) (2)(0)(1) ( 1)(2)(2)
= 5 + 20 + 75 + 4 = 104
x1 = 1det(A )
det(A) = 130
26 = 5, x2 = 2det(A )
det(A) = -52
26 = 2, x3 = 3det(A )
det(A) = 104
26 = 4
F F F ก ก
ʿก 3.2
Fก F กF ก F
1. x + 2y z = 0
2x + y + z = 3
x + y + 2z = 5
2. w x + y z = 4
4w x + 3y + z = 8
2w + x + y z = 0
3w + 2x + y 3z = 1

40. 34 ก F
3.2.5 ก กF ก F Fก ก
F F F ก กF ก F Fก F ˈ
F F ก ก ก กF ก F
Fก ก F ก ก ก ก F F 3.1 FกF
F ก กF ก F Fก ก F
ก ก ก F F ก ก กF ก F ก F F
F 3.6 ก F A =
1 2 1 0
-1 1 0 1
 
 
 
F A ˈ ก F F
ก กF ก F Fก ก F ก F
ก (A) F ˈ ก F F ก ก F F (B) F F
F
F 3.7 ก ก F A F 3.4 A =
3 2
1 1
 
 
 
F ก F F A
[A | B] =
3 2 1
1 1 0
 
 
 
F 3.8 ก ก F A F 3.5 A =
1 2 1
2 -5 -5
3 2 -3
 
 
 
 
 
F ก F F A
[A | B] =
1 2 1 5
2 -5 -5 0
3 2 -3 -1
 
 
 
 
 
3.5
ก F F (augmented matrix) ก F ก กก ก ก ก
กก F ⋮ |

41. F F 35
ก ก กF ก F Fก ก F F F
F Fก ก ก F F F [In | X] F F
F 3.9 ก F 3.8 F [A | B] =
1 2 1 5
2 -5 -5 0
3 2 -3 -1
 
 
 
 
 
ก
ก
1 2 1 5
2 -5 -5 0
3 2 -3 -1
 
 
 
 
 
2 2 1
3 3 1
R R - 2R
R R - 3R
→
→
→
1 2 1 5
0 -9 -7 -10
0 -4 -6 -16
 
 
 
 
 
1
1 1 32
2 2 3
R R + R
R R - 2R
→
→
→
1 0 -2 -3
0 -1 5 22
0 -4 -6 -16
 
 
 
 
 
3 3 2R R -4R→
→
1 0 -2 -3
0 -1 5 22
0 0 -26 -104
 
 
 
 
 
( )1
3 326R - R→
→
1 0 -2 -3
0 -1 5 22
0 0 1 4
 
 
 
 
 
2 2 3R R - 5R→
→
1 0 -2 -3
0 -1 0 2
0 0 1 4
 
 
 
 
 
( )2 2R -R→
→
1 0 -2 -3
0 1 0 -2
0 0 1 4
 
 
 
 
 
1 1 3R R + 2R→
→
1 0 0 5
0 1 0 -2
0 0 1 4
 
 
 
 
 
F ก F F F ˈ ก F [In | X] F
ก (x, y, z) = (5, 2, 4)
ʿก 3.2
กF ก ก F ʿก 3.2 Fก ก

42. 36 ก F
3.3 ก F ก ก F Fก ก
ก F ก F F กF ก F ก A 1
= adj(A)
det(A)
F ก F ก ก Fก ก ก ก F F
ก ก F F F
F ก F F ก F ก A F ก F F [A | In] ก Fก
ก ก F ก F [In | A 1
] F ก F ก ก กF Fก
ก F ก ก F ก F F F FกF F F F
ก F ก F F ก 0 ก F F det(A) = 0 F F F ก F
F F ก F F
F 3.10 ก F A =
3 2
1 1
 
 
 
F A F ก F F ก
F ก A
ก det(A) = (3)(1) (1)(2) = 1 ≠ 0 F F ก
F [A | I2] =
3 2 1 0
1 1 0 1
 
 
 
2 1 2R R - 3R→
→
3 2 1 0
0 -1 1 -3
 
 
 
1 1 2R R + 2R→
→
3 0 3 -6
0 -1 1 -3
 
 
 
1
1 13
2 1
R R
R (-1)R
→
→
→
1 0 1 -2
0 1 -1 3
 
 
 
F ก F F F [I2 | A 1
]
F ก A A 1
=
1 -2
-1 3
 
 
 
F 3.11 ก F A =
1 2 1
2 -5 -5
3 2 -3
 
 
 
 
 
F A F ก F F
ก F ก A ก F
ก det(A) = 26 ≠ 0 F A F ก
F [A | I3] =
1 2 1 1 0 0
2 -5 -5 0 1 0
3 2 -3 0 0 1
 
 
 
 
 

43. F F 37
2 2 1
3 3 1
R R - 2R
R R - 3R
→
→
→
1 2 1 1 0 0
0 -9 -7 -2 1 0
0 -4 -6 -3 0 1
 
 
 
 
 
2 2 3R R - 2R→
→
1 2 1 1 0 0
0 -1 5 4 1 -2
0 -4 -6 -3 0 1
 
 
 
 
 
3 3 2R R - 4R→
→
1 2 1 1 0 0
0 -1 5 4 1 -2
0 0 -26 -19 -4 9
 
 
 
 
 
( )1
3 326R - R→
→
19 94
26 26 26
1 2 1 1 0 0
0 -1 5 4 1 -2
0 0 1 -
 
 
 
 
  
2 2 3R R - 5R→
→ 9 6 7
26 26 26
19 94
26 26 26
1 2 1 1 0 0
0 -1 0 -
0 0 1 -
 
 
 
 
  
1 1 3R R - R→
→
7 94
26 26 26
9 6 7
26 26 26
19 94
26 26 26
1 2 0 -
0 -1 0 -
0 0 1 -
 
 
 
 
  
1 1 2R R + 2R→
→
25 8 5
26 26 26
9 6 7
26 26 26
19 94
26 26 26
1 0 0 -
0 -1 0 -
0 0 1 -
 
 
 
 
  
2 2R (-1)R→
→
25 8 5
26 26 26
9 6 7
26 26 26
19 94
26 26 26
1 0 0 -
0 1 0 - -
0 0 1 -
 
 
 
 
  

44. 38 ก F
F ก F F F ก [I3 | A 1
] A 1
=
25 8 5
26 26 26
9 6 7
26 26 26
19 94
26 26 26
-
- -
-
 
 
 
 
  
ʿก 3.3
1. F ก F ก F F 3.10 3.11 F ก F ก F
ˈ ก AA 1
= A 1
A = In
2. F A =
1 2 -1
2 1 1
1 1 2
 
 
 
 
 
F ก F F F ก A
F F F AA 1
= A 1
A = I3
3. F A =
3 2 -1 1
1 2 -2 3
0 1 -3 -1
2 5 -1 3
 
 
 
 
 
 
F ก F F F ก
A

45. F F 39
ก
ก F . ก F F ก ก F. ก : ,
2546.
. F Ent 47. ก : F F ก F, 2547.
. F. F ก. ก : ก F, 2533.