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If $\mathrm{A}$ and $\mathrm{B}$ are square matrices of second order such that $|\mathrm{A}|=-1,|\mathrm{~B}|=3$, then what is $|3 \mathrm{AB}|$ equal to?
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The correct answer is:
$-27$
We know that,$|\mathrm{kA}|=\mathrm{k}^{\mathrm{n}}|\mathrm{A}|$, where $\mathrm{n}$ is order of matrix $\mathrm{A}$.
$\therefore|3 \mathrm{AB}|=3^{2}|\mathrm{~A}||\mathrm{B}| \quad(\because|\mathrm{AB}|=|\mathrm{A}||\mathrm{B}|)$
$=9(-1)(3)$
$=-27 \quad(\because|\mathrm{A}|=-1,|\mathrm{~B}|=3)$
$\therefore|3 \mathrm{AB}|=3^{2}|\mathrm{~A}||\mathrm{B}| \quad(\because|\mathrm{AB}|=|\mathrm{A}||\mathrm{B}|)$
$=9(-1)(3)$
$=-27 \quad(\because|\mathrm{A}|=-1,|\mathrm{~B}|=3)$
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