Determinant of a portioned matrix

Question

If $A$ and $B$ are $n\times n$ matrices and $C$ is defined to be

$$ C=\begin{pmatrix} O&A\\ B&O \end{pmatrix} $$ Where $O$ denotes the zero matrix.

Can I conclude that $O$ needs to be only square matrix of size $n\times n$ because if $O$ has any other size, then it does not make sense?

Can I also then say that $\det(C)=-\det(A)\det(B)$?

If I am wrong, then can you explain why?

Answer 1

Consider the example where $A$ and $B$ are both the $2\times2$ identity matrix. Then you may calculate that $\det(C)$ is $1$, which is not $-\det(A)\det(B)$. In fact we have in general $$\det(C)=(-1)^n\det(A)\det(B)\ .$$ Reason: use row-reduction. If we interchange rows $n$ times we get $$\det(C)=(-1)^n\det\pmatrix{B&O\cr O&A\cr}\ ,$$ and it's not too hard to show that the matrix on the RHS has determinant $\det(A)\det(B)$.

Answer 2

Yes, in your situation $O$ has to be $n\times n$. As for the determinant, $$ \det C=(-1)^n\,\det(A)\,\det(B). $$ This is because $$ \begin{bmatrix}O&A\\ B& O\end{bmatrix}=\begin{bmatrix}O&I\\ I& O\end{bmatrix}\,\begin{bmatrix}B&O\\ O&A\end{bmatrix} $$