Let A, B, C be m x n matrices and p and q be two non-zero scalars (numbers). Then we have the following properties.
(a) Matrix multiplication is not commutative in general :
If A is of order m x n and B of the order n x p then AB is defined but BA is not defined. Even if AB and BA are both defined, it is not necessary that they are equal.
In general AB ≠ BA.
(b) Matrix multiplication is distributive over matrix addition :
(i) If A, B, C are m x n , n x p and n x p matrices respectively then
A(B + C) = AB + AC (Right Distributive Property)
(ii) If A, B, C are m x n, m x n and n x p matrices respectively then
(A + B)C = AC + BC (Left Distributive Property)
(c) Matrix multiplication is always associative :
If A, B, C are m x n , n x p and p x q matrices respectively then
(AB)C = A(BC)
(d) Multiplication of a matrix by a unit matrix :
If A is a square matrix of order n x n and I is the unit matrix of same order then
AI = IA = A
(I is called multiplicative identity)
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