Determinant of a matrix in a block form

Question

Let $A, B, C$ be matrices with size $m \times m$, $n \times n$, and $n \times m$, respectively. If $\det(A) = 2$ and $\det(B) = 3,$ then find $$\det \begin{pmatrix} 0 & A \\ B & C \end{pmatrix} =\ldots $$

I stuck to solve this problem. I also wonder how can we calculate a determinant of matrix with some matrices in it (submatrices)? Please, anyone help me

Answer 1

Hints.

Step 1. $$ \det \left(\begin{array}{cc} 0 & A \\ B & C\end{array}\right) =(-1)^m\det \left(\begin{array}{cc} A & 0 \\ C & B\end{array}\right) $$

Step 2. $$ \det \left(\begin{array}{cc} A & 0 \\ C & B\end{array}\right)=\det A\cdot \det B $$

Step 1, is obtained by $m^2$ permutations of rows and as many changes of sign.

Step 2, is obtained using the Jordan forms of $A$ and $B$.