Question

If A be any $m\times n$ matrix and both AB and BA are defined prove that B should be $n\times m$ matrix

Answer 1

Since A is $m\times n$ and AB defined therefore B should be $n\times p$ because the number of columns of A should be equal to number of rows of B.
Again B is row $n\times p$ and A is $m\times n$.
And since BA is also defined therefore p should be equal to m by the same argument as above i.e., p = m
$\therefore$ B is $n\times m$ matrix.