Answer to Question #156145 in Linear Algebra for Lea

Question

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 ).

Answer 1

Using the multiplicativity of the determinant, we have:

$\det B^{-1}\cdot\det B = \det(BB^{-1}) = \det E = 1$ , therefore, $\det B^{-1} = (\det B)^{-1}$

$\det (BAB^{-1}) = \det B\cdot\det A\cdot\det B^{-1} = \det A$

$\det(\lambda A) = \det(\lambda E\cdot A)=\det(\lambda E)\det A = \lambda^4\det A$

Moreover, $\det A^T = \det A$.

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