Under what conditions is $AA^T$ invertible?
Let $D=A'$ so that $B=D'D$. The necessary and sufficient condition for the invertibility of $B$ is that $D$ has full column rank (i.e. $A$ has full row rank).
Necessary: suppose that $D'D$ is invertible and suppose that there exists some nonzero $x$ such that $Dx=0$. But then $Dx=0$ implies $D'Dx=0$ violating the nonsingularity of $D'D$.
Sufficient: because $D'D$ is square, it's enough to show that it has full column rank. Suppose not: there is some nonzero $x$ such that $D'Dx=0$. But this implies $0=x'D'Dx=|Dx|^2$, that is $Dx=0$, violating $D$ having full column rank.