I found the following formula on Wikipedia:
Suppose $A$, $B$, $C$, and $D$ are matrices of dimension $n \times n$, $n \times m$, $m \times n$, and $m \times m$, respectively.
$$\det \begin{pmatrix} A & B \\ C & D\end{pmatrix} = \det(D) \det \left(A – B D^{-1} C \right)$$
Hence, if $D$ is a zero matrix, the determinant should be zero. But when I calculate the determinant of the following matrix
$$\det \left[ \begin {array}{ccc|ccc} 0&0&0&-4&0&0\\ 0&-1&0
&0&-5&0\\ 0&0&-2&0&0&-6\\ \hline 7&0&0&0&0
&0\\ 0&8&0&0&0&0\\ 0&0&9&0&0&0
\end {array} \right] = 60480
$$
Is this formula above really only applicable, when $n\neq m$?
Best Answer
Since all four blocks are square and the bottom ones do commute (the zero matrix commutes with everyone),
$$\det \left[ \begin {array}{ccc|ccc} 0&0&0&-4&0&0\\ 0&-1&0 &0&-5&0\\ 0&0&-2&0&0&-6\\ \hline 7&0&0&0&0 &0\\ 0&8&0&0&0&0\\ 0&0&9&0&0&0 \end {array} \right] = \det \left( \mathrm O_3 + \mbox{diag} (4 \cdot 7, 5 \cdot 8, 6 \cdot 9) \right) = \prod_{i=4}^9 i = \frac{9!}{3!} = \color{blue}{60480}$$