Question

Show that matrix multiplication is not commutative.

Short answer

Matrix multiplication is not commutative.

Step-by-step solution

Step: 1

Let$\mathrm{A}=[{\mathrm{a}}_{{\mathrm{i}}_{\mathrm{j}}}{]}_{\mathrm{m}\times \mathrm{n}}$$B=[{b_i}_j{]}_{m\times n}$

Matrix multiplication XY is valid when, No. of columns in pre-multiplier

(X)=No. of rows in post-multiplier (Y)

$X_{a\times b}{Y}_{b\times c}\Rightarrow (XY{)}_{a\times c}$

Step: 2

${\mathrm{A}}_{\mathrm{m}\times \mathrm{n}}{\mathrm{B}}_{\mathrm{n}\times \mathrm{p}}$

No of columns in A= n=No of rows in B

AB is valid and we can evaluate it.

No of columns in B=P

No of rows in A=m

$p\ne m$

BA is not valid.

Step: 3

If p=m=n then and then only AB and BA will have the same order. AB may or may not be equal to BA.