Mathematics

Problem

Let

A

be an

m\times n

matrix,

B

n\times p

matrix and

C

p\times q

matrix where

m,n,p,q\in\mathbb{N}

. (i) Determine the orders of

AB,\;BC

and

ABC

. (ii) Show that the product

ACB

may fail to exist. Obtain the necessary and sufficient condition on

m,\,n,\,p,\,q

for

ACB

to be defined. (iii) If both

ABC

and

ACB

are defined simultaneously, prove that

n=p