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Properties of Matrix Multiplication

Let A, B and C be matrices of dimensions such that the following are defined.  Then
Associative:
A(BC)  =  (AB)C
Distributive: A(B + C)  =  AB + AC
Distributive: (A + B)C = AC + BC
c(AB) = (cA)B = A(cB), where c is a constant, please notice that A∙B ≠ B∙A

Multiplicative Identity: For every square matrix A, there exists an identity matrix of the same order such that IA = AI =A.

Example 1: Verify the associative property of matrix multiplication for the following matrices.

Solution: Here we need to calculate both R.H.S (right-hand-side) and L.H.S (left-hand-side) of A (BC) = (AB) C using (associative) property.

On the RHS we have:

and

On the LHS we have:

and

Hence the associative property is verified.

Example 2: Verify the distributive property of matrix multiplication for the following matrices.

Solution: Here we need to calculate both R.H.S and L.H.S of A(B+C) = AB+AC (distributive) property.

Hence the property is verified.

Example 3: Find (AB) C and A (BC).

(AB)C is determined as:

A(BC) is determined as: