Properties of Matrix Multiplication
(1) Matrix multiplication is not commutative in general. For matrices A and B , we need
not have AB = BA .
For example:
(a) A and B are 2 x 3 and 3 x 4 matrices, then AB is a 2 x 4 matrix whereas BA is
not defined.
(b) A and B are 2 x 3 and 3 x 2 matrices, then AB is a 2 x 2 matrix and BA is a
3 x 3 matrix.
(c) A and B are 2 x 2 matrices, then both AB and BA are 2 x 2 matrices. Even in this
case, we may not have AB = BA .
(2) Associative Law of Multiplication : If A , B and C be matrices of the type m x n , n x p
p x q respectively, then (AB)C = A(BC) .
(3) Matrix Multiplication is Distributive with respect to Addition :
(a) If A , B and C are matrices of the type m x n , n x p and n x p respectively, then
A(B + C) = AB + AC
(b) If A , B and C are matrices of the type m x n , m x n and n x p respectively, then
(A + B)C = AC + BC
(4) The Product of non-zero matrices may be a zero matrix. For matrices A and B , it may
be possible that AB = O and neither A nor B is a zero matrix.
Comments
Post a Comment