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Given an $m \times m$ square matrix $M$:

$$ M = \begin{bmatrix} A & 0 \\ 0 & B \end{bmatrix} $$

$A$ is an $a \times a$ and $B$ is a $b \times b$ square matrix; and of course $a+b=m$. All the terms of A and B are known.

Is there a way of calculating determinant of $M$ by determinants of (or any other useful data from) of $A$ and $B$ sub-matrices?

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Hint: It's easy to prove that $$\det M=\det A\det B$$

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