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Linear Algebra

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Moderate

Determinants of Block Matrices

LINALG-KYJ4@@

If $A=\begin{bmatrix} a_{11} & a_{12} \\\ a_{21} & a_{22} \end{bmatrix}$ and $B=\begin{bmatrix} b_{11} & b_{12}\\\ b_{21} & b_{22}\end{bmatrix}$ and $C=\begin{bmatrix} a_{11} & a_{12} & 0 & 0 \\\ a_{21} & a_{22} & 0 & 0\\\ 0 & 0 & b_{11} & b_{12} \\\ 0 & 0 & b_{21} & b_{22}\end{bmatrix}$ then $det(C)$ equals

A

$det(AB)$

B

$det(AB-BA)$

C

$det(A)^2+det(B)^2$

D

$det(A)^{det(B)}$