# A,B,C are 3 times 3 matrices with det(A)=-3 , det(D)=-2, det(C)=6.What is det(A^2BC^{-1})? -3 -18 18 -108 36

A,B,C are $3×3$ matrices with det(A)=-3 , det(D)=-2, det(C)=6.What is $det\left({A}^{2}B{C}^{-1}\right)$?
-3
-18
18
-108
36
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Delorenzoz
Step 1
Given A, B and C are $3×3$ matrices.
and
$|A|=-3$
$|B|=-2$
$|C|=6$
Find the determinant of $\left({A}^{2}B{C}^{-1}\right)$
Step 2
since,
$|AB|=|A|\cdot |B|$
$|{A}^{n}|=|A{|}^{n}$
and
$|{A}^{-1}|=\frac{1}{|A|}$
Then,
$|{A}^{2}B{C}^{-1}|=|{A}^{2}||B||{C}^{-1}|$
$=|{A}^{2}||B|×\frac{1}{|C|}$
Substitute all the given values,
Step 3
$|{A}^{2}B{C}^{-1}|=\left(-3{\right)}^{2}\left(-2\right)×\frac{1}{6}$
$=9×\left(-2\right)×\frac{1}{6}$
$=-18×\frac{1}{6}$
$=-3$
Hence,
$|{A}^{2}B{C}^{-1}|=-3$