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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.

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    $\begingroup$ For small enough $\varepsilon>0$, you will have that $D+tI$ will be invertible for all $t\in(0,\varepsilon)$. So the determinant identity will hold when $D$ is replaced by $D+tI$. Then use a continuity argument. $\endgroup$ – Ewan Delanoy Sep 29 '16 at 14:59
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    $\begingroup$ You can see here: ee.iisc.ac.in/people/faculty/prasantg/downloads/blocks.pdf $\endgroup$ – Emilio Novati Sep 29 '16 at 15:40

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