PROPERTIES OF MATRIX MULTIPLICATION

(i) Commutative Property :

If A and B are two matrices and if AB and BA both are defined, it is not necessary that

AB  =  BA

Note :

Multiplication of two diagonal matrices of same order is commutative. Also, under matrix multiplication unit matrix commutes with any square matrix of same order.

(ii) Associative Property :

For any three matrices A, B and C, we have

(AB)C  =  A(BC)

whenever both sides of the equality are defined.

(iii) Matrix multiplication is distributive over addition :

For any three matrices A, B and C, we have

(i) A(B + C)  =  AB + AC

(ii) (A + B)C  =  AC + BC

whenever both sides of equality are defined

(iv)  Existence of multiplicative identity :

For any square matrix A of order n, we have

AI  =  IA  = A

where I is the unit matrix of order n. Hence, I is known as the identity matrix under multiplication.

(v)  Existence of multiplicative inverse :

If A is a square matrix of order n, and if there exists a square matrix B of the same order n,

such that AB  =  BA  =  I

where I is the unit matrix of order n, then B is called the multiplicative inverse matrix of A

It is denoted by A^-1. 

(vi) Reversal law for transpose of matrices :

If A and B are two matrices and if AB is defined,

then

(AB)^T  =  B^TA^T

Matrix Multiplication is not Commutative - Example

Find the product of A and B : 

AB  =

1 2 5 3

x

2 5 7 3

=

1 2

x

2 7

1 2

x

5 3

5 3

x

2 7

5 3

x

5 3

=	(2+14) (5+6) (10+21) (25+9)

AB  =

16 11 31 34

Find the product of B and A : 

BA  =

2 5 7 3

x

1 2 5 3

=

2 5

x

1 5

2 5

x

2 3

7 3

x

1 5

7 3

x

2 3

=	(2+25) (4+15) (7+15) (14+9)

BA  =

27 19 22 23

Therefore, 

AB  ≠  BA

Associative Property - Example

Find the product of B and C :

BC  =

2 5 7 3

x

1 3 5 1

=

2 5

x

1 5

2 5

x

3 1

7 3

x

1 5

7 3

x

3 1

=	(2+25) (6+5) (7+15) (21+3)

BC  =

27 11 22 24

Find the product of A and (BC) :

A(BC)  =

1 2 5 3

x

27 11 22 24

=	(27+44) (11+48) (135+66) (55+72)

A(BC)  =

71 59 201 127

Find the product of A and B :

AB  =

1 2 5 3

x

2 5 7 3

=

1 2

x

2 7

1 2

x

5 3

5 3

x

2 7

5 3

x

5 3

=	(2+14) (5+6) (10+21) (25+9)

AB  =

16 11 31 34

Find the product of (AB) and C :

(AB)C  =

16 11 31 34

x

1 3 5 1

=

16 11

x

1 5

16 11

x

3 1

31 34

x

1 5

31 34

x

3 1

=	(16+55) (48+11) (31+170) (93+34)

(AB)C  =

71 59 201 127

Therefore, 

A(BC)  =  (AB)C

Distributive Property - Example

Find the addition of B and C :

B+C  =

2 5 7 3

+

1 3 5 1

=

3 8 12 4

Find the product of A and (B + C) :

A(B + C)  =

1 2 5 3

x

3 8 12 4

=

1 2

x

3 12

1 2

x

8 4

5 3

x

3 12

5 3

x

8 4

=	(3+24) (8+8) (15+36) (40+12)

A(B + C)  =

27 16 51 52

Find the product of A and B :

AB  =

1 2 5 3

x

2 5 7 3

=

1 2

x

2 7

1 2

x

5 3

5 3

x

2 7

5 3

x

5 3

=	(2+14) (5+6) (10+21) (25+9)

AB  =

16 11 31 34

Find the product of A and C :

AC  =

1 2 5 3

x

1 3 5 1

=

1 2

x

1 5

1 2

x

3 1

5 3

x

1 5

5 3

x

3 1

=	(1+10) (3+2) (5+15) (15+3)

AC  =

11 5 20 18

Find the addition of AB and AC :

AB + AC  =

16 11 31 34

+

11 5 20 18

=	(16+11) (11+5) (31+20) (34+18)

AB + AC  =

27 16 51 52

Therefore,

A(B + C)  =  AB + AC

Identity Property - Example

Find the product of A and I :

AI  =

1 2 5 3

x

1 0 0 1

=

1 2

x

1 0

1 2

x

0 1

5 3

x

1 0

5 3

x

0 1

=	(1+0) (0+2) (5+0) (0+3)

AI  =

1 2 5 3

Find the product of I and A :

AI  =

1 0 0 1

x

1 2 5 3

=

1 0

x

1 5

1 0

x

2 3

=

0 1

x

1 5

0 1

x

2 3

=	(1+0) (2+0) (0+5) (0+3)

IA  =

1 2 5 3

Therefore,

AI  =  IA

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