Let $A$ be an $n\times m$ real matrix, and let $B$ be an $n\times(n-m)$ real matrix, please prove this matrix inequality. $A^{\prime}$ represents the transpose of the matrix. $$ \det\begin{bmatrix} A^{\prime}A&A^{\prime}B\\ B^{\prime}A&B^{\prime}B\\ \end{bmatrix} \le \det(A^{\prime}A)\det(B^{\prime}B) $$
This problem comes with a constraint, which is to primarily utilize basic matrix operations. My approach is to decompose the left matrix into the product of several other matrices, I've been thinking for a long time, but I have no other ideas.