Question

Let A and B be 3x3 matrices with det(A) = -2 and det(B) = 6. Which of the following evaluations are correct?

I. det(A³) = -8
III. det(A⁻¹B) = -3
IV. det(Bᵀ²) = 1/6

Select one:
a-I, III, and IV
b-II, III, and IV
c-II and III
d-I and III
e-I and II

Answer 1

The correct evaluations are I and III, as det(A³) equals -8 and det(A⁻¹B) equals -3. Evaluation IV is incorrect because det(Bᵀ²) should be 36, not 1/6. The answer to the student's question is options I and III are correct.

Step-by-step explanation:

Let's address the evaluations based on the properties of determinants for the given 3x3 matrices A and B, with det(A) = -2 and det(B) = 6.

I. det(A³) can be found using the rule that det(A³) = [det(A)]³. Therefore, det(A³) = (-2)³ = -8.

III. det(A⁻¹B) requires us to know the property that det(AB) = det(A)det(B), and also that det(A⁻¹) = 1/det(A). Thus, det(A⁻¹B) = det(A⁻¹)det(B) = (-1/2)(6) = -3.

IV. det(Bᵀ²) is incorrect because the determinant of a square matrix raised to a power is the determinant of the matrix raised to that power. Hence, det(Bᵀ²) = [det(B)]², which is 6² = 36, not 1/6.