I have a question about the problem below.
Let $A$, $B$, $C$, $D$, be commuting $n \times n$ matrices over the field $F$. Show that the determinant of the $2n \times 2n$ matrix $\begin{bmatrix} A & B \\ C & D \\ \end{bmatrix}$ is $det(AD - BC)$.
I've proven this to be true if $D$ is invertible, but now I need to show that I can reduce to that case. Any hints would be much appreciated.