Properties of transpose matrices

Question

It's known that if $A$ is a matrix such that $A.A^T = 0$ then $A = 0$

Suppose I let $A = (a_{ij} )$ be $m×n$ matrix and $A^T = (a_{ji})$ be its transpose.

However, I do not understand how do i go about proving the quoted concept above.

Answer 1

Hint: If $A$ has rows $r_1=(a_{1,1},a_{1,2},...,a_{1,n}),...,r_m=(a_{m,1},a_{m,2},...,a_{m,n})$, then $AA^T$ has diagonal entries $r_1 \cdot r_1, r_2 \cdot r_2,...,r_m \cdot r_m.$