Question #156145

Suppose that A and B are 4 × 4 matrices such that B is non-singular. Which of the following statement(s) is/are correct?

1. det(−3A T ) = 34 det(A) and det(A) = det(BAB−1 ).

2. det(3B −1 ) = 34 det(B) and det(A) = det(BAB−1 ).

3. det(2A T ) = −16 det(A) and det(A) = det(BAB−1 ).

4. det(2A T ) = 23 det(A) and det(A) = det(BAB−1 ).


1
Expert's answer
2021-01-19T04:00:57-0500

Using the multiplicativity of the determinant, we have:

detB1detB=det(BB1)=detE=1\det B^{-1}\cdot\det B = \det(BB^{-1}) = \det E = 1 , therefore, detB1=(detB)1\det B^{-1} = (\det B)^{-1}

det(BAB1)=detBdetAdetB1=detA\det (BAB^{-1}) = \det B\cdot\det A\cdot\det B^{-1} = \det A

det(λA)=det(λEA)=det(λE)detA=λ4detA\det(\lambda A) = \det(\lambda E\cdot A)=\det(\lambda E)\det A = \lambda^4\det A

Moreover, detAT=detA\det A^T = \det A.

From this we have that statement 1 is correct, but statements 2-4 are not.


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