Let $X$ be an $n \times p$ matrix with $\text{rank}(X) \le p$. Let $A$ be a $n \times n$ positive definite matrix.
When $\text{rank}(X) =p$, $X$ is of full rank, and $|X'AX|>0$ should hold. Similarly, when $\text{rank}(X) < p$, $|X'AX|=0$ should hold. However, I don't know how to prove these statements.
Also, I'm wondering if $|X'AX|>0$ implies that $\text{rank}(X) = p$ and if $|X'AX|=0$ implies that $\text{rank}(X) < p$.
Any comments would be appreciated!