Let $A$ be the positive definite matrix. I used the inverse of $X^{\prime}AX$ somewhere in my calculations, for some real-valued matrix $X$, and I set the assumption that $X^{\prime}AX$ is a nonsingular matrix. I also argue that a sufficient condition is that $X$ is a full rank matrix.
I would like to know if this condition is also necessary; that is, $X^{\prime}AX$ is a nonsingular matrix if and only if $X$ is a full rank matrix.