Properties of Matrix Multiplication

(i) Matrix multiplication is associative

i.e. (AB)C = A(BC), if A, B, C are m × n, n × p, p × q matrices respectively.

(ii) Multiplication of matrices is distributive over addition of matrices.

i.e.,  A(B + C) = AB + AC

(iii) Existence of multiplicative identity of  square matrices.

If A is a square matrix of order n and In is the identity matrix of order n,

then A In = In A = A.

(iv) Whenever AB and BA both exist it is not necessary that AB = BA.

Thus AB ≠ BA

(v) The product of two matrices can be a zero matrix while neither of them is a zero matrix.

e.g. If A =  while neither A nor B is a
null matrix.

(vi) In the case of matrix multiplication if AB = 0, then it doesn't necessarily imply that

A = 0 or B = 0 or BA = 0.