The inverse of a 2x2 matrix is given by: A-1 = (1/det(A)) * [d -b] [-c a] Where det(A) is the determinant of A, which is ad - bc. This formula works for any invertible 2x2 matrix.

3. A matrix is an ordered rectangular array of
numbers. The size of a matrix is given by the
number of rows and the number of columns.
Let m denote the number of rows
Let n denote the number of columns.
Let aij denote the entry in the ith row and the
jth column.

4. m n
a11 a12 a1n
a21 a22 a2 n
m rows
am1 am 2 amn
n columns

5. Zero matrix
ij
square matrix

7. The Identity Matrix, denoted I n , is a
diagonal matrix of order nxn with all the
diagonal entries equal to 1.

8. 0 0 0 d11 0 0
0 0 0 0 d 22 0
O D
0 0 0 0 0 d mm
1 0 0
0 1 0
I Let I n denote an n n identity matrix
0 0 1

9. A square matrix all its
elements below the
main diagonal are
zeros. 0 i j
aij
0 i j
2 3 3
A 0 7 1
0 0 9

10. A square matrix all its
elements above the
main diagonal are
zeros. 0 i j
aij
0 i j
2 0 0
A 5 7 0
2 8 9

11. The matrix A is 3x3 of the form
2i i>j
aij
i j i j
a11 a12 a13 2 3 4
A 4 4 5
A a21 a22 a23
6 6 6
a31 a32 a33

12. 2i i j
aij 5 i j
i j i j
3j i j
bij
i j i j
If A and B are 3x3 find A-2B

13. a11 a12 a13 5 1 2
A a 21 a 22 a 23 4 5 1
a 31 a 32 a 33 6 6 5
3 3 4
B 3 6 5
3 6 9
1 7 10
A 2B 2 7 11
0 6 13

14. Equality of Matrices:
Two matrices are equal if they have the
same size and their corresponding entries
are equal.

15. Find x and y that satisfies the following
equation
x y y y
y
y=2
x+6=2 x=-4

16. A new matrix C may be defined as the
additive combination of matrices A and
B where: C = A + B
is defined by:
ij ij ij
i 1, 2,..., m and j 1, 2,..., n
Note: Only matrices of the same dimension can
be added

17. Addition of Matrices
a11 a12 a1n b11 b12 b1n
a21 a22 a2 n b21 b22 b2 n
A and B
am1 am 2 amn bm1 bm 2 bmn
a11 b11 a12 b12 a1n b1n
a21 b21 a22 b22 a2 n b2 n
A B
am1 bm1 am 2 bm 2 amn bmn

18. 3 4 1 2 4 6
A B C
5 6 3 4 8 10

19. Multiplying a matrix by a real number (scalar) results
in a matrix with each entry multiplied by the scalar. Let
k be a real number.
a11 a12 a1n ka11 ka12 ka1n
a21 a22 a2 n ka21 ka22 ka2 n
A kA
am1 am 2 amn kam1 kam 2 kamn

21. k(A + B) = kA + k B
(x+y) A = x A + y A
(x y) A = x ( y A)
oA=O
xO=O

22. C = A - B
is defined by
C = A + (-1) B
a11 b11 a12 b12 a1n b1n
a21 b21 a22 b22 a2n b2n
A B
am1 bm1 am2 bm2 amn bmn

23. The scalar matrix can be written as .
C= In
x 0 0 0
0 x 0 0
0 0 x 0
0 0 0 x

24. Matrices A and B have these dimensions:

25. Matrices A and B can be multiplied if:

26. The resulting matrix will have the dimensions:

27. a11 a12 a1n b11 b12 b1 p
a21 a22 a2 n b21 b22 b2 p
A and B
am1 am 2 amn bn1 bn 2 bnp
c11 c12 c1 p
Entry cij is obtained by
c21 c22 c2 p taking the sum of the
AB C
products of the entries of
cm1 cm 2 cmp the ith row in A with the
jth column in B.
Entry c11 a11b11 a12b21 ... a1nbn1
c21 a21b11 a22b21 ... a2 nbn1
c22 a11b12 a12b22 ... a1nbn 2

28. mxn mxs sxn
s
c ij aik b kj
k 1
i 1, 2,..., m
j 1, 2,..., n

29. 2 3
111
A 1 1 and B
1 0 2
1 0
[3 x 2] [2 x 3]
A and B can be multiplied
[3 x 3]

30. 2*1 3*1 5 2*1 3*0 2 2*1 3*2 8 5 2 8
C 1*1 1*1 2 1*1 1*0 1 1*1 1*2 3 2 1 3
1*1 0*1 1 1*1 0*0 1 1*1 0*2 1 1 1 1

31. 2x3 3x2 2x2
2 3
1 1 1
BxA 1 1
1 0 2
1 0
4 4
=
4 3

32. Properties of Matrix Multiplication
1.AB BA
2- AB C A BC
3. A (B C ) AB AC
(B+C)A=BA+CA
4. If A is an m n matrix and I m and I n are
m m and n n identity matrices, respectively,
then
I m A AI n A

33. 5- Amxn 0nxk =0mxk
0kxm Amxn = 0kxn

35. If A is an m x n matrix with elements aij, then the
transpose of A, denoted, AT, is an n x m matrix
with elements aji.
a11 a12 a1n a11 a21 am1
a21 a22 a2 n T
a12 a22 am 2
A A
am1 am 2 amn a1n a2 n amn

36. T T
T T T
T T T
T T

37. A square matrix B is said to be symmetric if
B = BT
brs = bsr s r
r=1,2, ,m
s=1,2, ,n

38. Any diagonal matrix is symmetric
A squared zero matrix is symmetric
The transpose of an upper triangular matrix is
a lower triangular matrix

39. A squared matrix is
said to skew-symmetric
if
A = - AT

40. 0 1 2
A 1 0 3
2 3 0
0 1 2
T
A 1 0 3
2 3 0
T
A A

41. The common term of a skew-symmetric
matrix can be written as
r s
ars
asr r s

42. If the common element of a 3x3 matrix is
given by
rs
Show that A is a skew-symmetric matrix

43. 0 1 1
A 1 0 1
1 1 0
0 1 1
T
A 1 0 1
1 1 0
T
A A

44. X =(x1,x2, .,xn) 1xn
AT =(1,1, .,1) 1xn
xi = X A = AT XT
xi2 = X XT

45. A square matrix is said to be orthogonal if
T T
T H E I D E N T I T Y M AT R I X I S
O R T H O G O N AL AN D S YM M E T R I C

46. -1 -1
2 2
A =
1 -1
2 2
1 -1 -1
A =
2 1 -1
T 1 -1 1
A =
2 -1 -1

47. T T
2

48. If A is a square matrix and k is a positive
integer, the power k of A is defined as
Ak = A . A ..A
So, A2 = A . A
and A0 = I

49. A square matrix A is said to be idempotent if
Ak = A for any positive integer k
The identity matrix is idempotent
The square zero matrix is idempotent

50. Show that the matrix A
defined below is
idempotent 1 1
2 2
A
1 1
2 2
1 1 1 1 1 1
2 2 2 2 2 2 2
A AA A
1 1 1 1 1 1
2 2 2 2 2 2
3
A A 2A AA A

51. Assume Ak-1 = A
Ak = Ak-1 A =A A=A
So Ak = A for any positive integer k
i.e. A is idempotent

52. Definition: Let A be an n n matrix. An inverse of A
is an n n matrix B such that:
AB = In and BA = In
A is then called invertible.
B is called an inverse of A
If A has no inverse, it is called singular.
.

53. The inverse (if exists) is unique.
Proof:
Assume A is invertible, with two inverses B and C, i.e.
AB=BA=I
and
AC = CA = I,
(BA)C=B(AC)=BI=B
(BA)C=IC=C
Thus, B=C

54. Since the inverse ,if it exists, is unique, we
call it A-1
A A-1 = A-1 A = In

55. Let A and B be invertible matrices of the same size,
and k be a nonzero scalar. Then:
1. I 1 =I
2. (A 1) 1 = A
3. (kA) 1 = k 1A 1

56. 4- (AB) 1 = B 1A 1
5- (An) 1 = (A 1)n
6- (AT) 1 = (A 1)T
7- If A is orthogonal then it is invertible and
A-1=AT

57. If A and B are nxn invertible matrices, prove
that AB is invertible also.
(AB)(AB)-1=(AB)(B-1A-1) = A(BB-1)A-1
-1 -1
n n
(AB)-1 (AB) = (B-1A-1) (AB)= B(AA-1)B-1
-1 -1
n n

58. Find the inverse of the 2 2 matrix:

59. 1
1

60. 1

61. 3 1
If A
5 2
Show that
A2 -5A+I2=0
Using this result find A-1

62. A2 - 5A + I2 =0
A-1 A A 5 A-1 A + A-1 = 0
A 5 I + A-1 = 0
A-1 =5I A
-1

63. 3 5
If A =
2 3
1 4 7
B
2 4
Fin d a m a trix X su ch th a t
XB =A

64. XB=A
X B B-1 = A B-1
X I = A B-1

65. Let A, B, and X be 3 invertible matrices of
the same size. Solve the following matrix
equation for X:
(A 1XB) 1 = (BA)2
Note: Be careful with the order of the matrix
multiplication.
Answer: X = (B2AB) 1 = B 1A 1(B 1)2

66. (A-1 X B)-1 = (B A)2
B-1 X-1 (A-1 )-1 = (B A)2
B-1 X-1 A = (BA)2
B B-1 X-1 A = B (BA)2
I X-1 A A-1 = B (BA)2 A-1
X-1 I = B BA BA A-1
X-1 = B2 A B
X = ( B2 A B)-1

67. Show that the inverse of the general 2 2 matrix:
a b
A
c d
1 d b
A
ad bc c a
1
1

68. -1 x y
A s s u m e A =
w z
-1 a b x y 1 0
A A = =
c d w z 0 1
a x + b w = 1
c x + d w = 0
-c a x - c b w = -c
c a x + a d w = 0
-c
w = a d - b c 0
a d - b c
d
x = a d -b c 0
ad b c