 # If A and B are 4 times 4 matrices, det(A) = 1, det(B) = 4, then det(AB) = ? det(3A) = ? det(A^T) = ? det(B^{-1}) = ? det(B^4) = ? Caelan 2020-11-07 Answered
If A and B are $4×4$ matrices, det(A) = 1, det(B) = 4, then
det(AB) = ?
det(3A) = ?
$det\left({A}^{T}\right)=?$
$det\left({B}^{-1}\right)=?$
$det\left({B}^{4}\right)=?$
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Given that
A and B are $4×4$ matrices:
det(A)=1 : det(B)=4 Then
(1) $det\left(AB\right)=det\left(A\right)\cdot det\left(B\right)$
$=1\cdot 4$
det(AB)=4
(2) $det\left(3A\right)=3\cdot det\left(A\right)$
$=3\cdot \left(1\right)$
det(3A)=3
(3) $det\left({A}^{T}\right)=det\left(A\right)$
$det\left({A}^{T}\right)=1$
(4)$det\left({B}^{-1}\right)=\frac{1}{|B|}$
$det\left({B}^{-1}\right)=\frac{1}{4}$
(5) $det\left({B}^{4}\right)=det\left(B\cdot B\cdot B\cdot B\right)$
$=det\left(B\right)\cdot det\left(B\right)\cdot det\left(B\right)\cdot det\left(B\right)$
$=4\cdot 4\cdot 4\cdot 4$
=256
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