Question

Prove that if the product ab is a square matrix, then the product ba is defined.

Answer 1

If A*B is defined then matrix A must have the same number of columns as B has rows. In other words,
dimensions of matrix A = m x n
dimensions of matrix B = n x p

So for now, matrix AB is m x p. Note how the n terms match up. The 'n' terms are the inner terms

(m x n) x (n x p)

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We're told that A*B is a square matrix, so that means m = p. We have the same number of rows and columns. This means

dimensions of matrix A = m x n
dimensions of matrix B = n x m

(m x n) x (n x m)

So matrix A*B is an m x m matrix.

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If we swap things around to compute B*A, then we can see that this is possible. Why? Because the 'm's now match up

dimensions of matrix B = n x m
dimensions of matrix A = m x n

The 'm's are now the inner terms. 

(n x m) x (m x n)

meaning that matrix B*A is an n x n matrix. This proves that B*A is defined.