Determine the typical (j kth) entry in A*A^T and the typical entry in A^T * A
Explain why it follows that if A*A^T or A^T * A is the zero matrix, then A = 0
Since we are able to multiple $\displaystyle A A^t$ and $\displaystyle A^t A$ then $\displaystyle A$ is square. If you look at the diagonal elements of say $\displaystyle A A^t$ they are of the form
$\displaystyle A A^t_{ii} = \sum_{j=1}^n a_{i j}^2$
and since $\displaystyle A A^t = 0$ then
$\displaystyle \sum_{j=1}^n a_{i j}^2 = 0\;\;\Rightarrow\;\;a_{ij} = 0\;\; \forall i,j$