Suppose is an matrix and is an matrix with

MATHEMATICS · Subjective · AP · AP Precalculus · Functions with Parameters, Vectors, and Matrices · Functions with Parameters, Vectors, and Matrices

Problem

Suppose

A

is an

m\times n

matrix and

B

is an

n\times m

matrix with

m\neq n

. (i) Show that both products

AB

and

BA

are defined and state their respective orders. (ii) Prove that if for two matrices

P

and

Q

both

PQ

and

QP

are defined, then either: (a)

P

and

Q

are square matrices of the same order, or (b) one of them is of order

m\times n

and the other is of order

n\times m

for some positive integers

m

and

n

. (iii) Give an explicit example of non-square matrices

P

and

Q

satisfying condition (b) in part (ii).