Conditions for invertibility of $AA^t$

Question

Let $A$ be a matrix whose rows are pmfs (i.e. nonnegative entries, each row sums to $1$). Are there any conditions on $A$ weaker than invertibility such that $AA^t$ is invertible?

Answer 1

In general, $AA'$ is invertible iff $A$ has full row rank.

For convenience, let $B=A'$, then we are to investigate $B'B$, which is positive semidefinite. Then, $$ B'B\text{ is invertible}\iff B'B\text{ is positive definite}\\ \iff\forall v\neq 0, 0<v'B'Bv=|Bv|^2\\ \iff\forall v\neq 0, Bv\neq 0\iff B\text{ has full column rank.} $$ (Every $v$ above is a column vector with length equals to the number of columns of $B$.)