If A is any mxn such that AB and BA are both defined show that B is an mxn matrix?

By rule of matrix multiplication the number of rows in the first matrix must equal the number of rows in the second matrix. If A is an axb matrix and B is a cxd matrix, then a = d. Then if BA is defined, then c = b. This means that B is not necessarily mxn, but must be nxm.