Prove that if A is m times n and B is n times m such that AB

Prove that if A is m times n and B is n times m such that AB = I_m and BA = I_n, then m = n.

Solution

Assumptions: AB = BA

Need to show: A and B are both square.

Let A be m × n, and B be p × q.

Since AB is defined, n = p.

Since BA is defined, q = m.

Therefore, we have that B is n × m. T

hus, AB is m × m BA is n × n Since those are equal,

we must have m = n. Thus, A and B are both n × n and hence are square, as required.

This is a correct proof !