Question

If \( A \) is a \(3 \times 3\) matrix with eigenvalues \(1, 2, 3\), what is the determinant of \(A^2\)?

• A. \(6\)
• B. \(12\)
• C. \(18\)
• D. \(36\)

Hint:

Two useful results:

• Product of eigenvalues = determinant
• \( \det(A^k) = (\det A)^k \)

Answer 1

The Correct Option is D

Concept 1: Determinant and eigenvalues
The determinant of a matrix is equal to the product of its eigenvalues: \[ \det(A) = \lambda_1 \cdot \lambda_2 \cdot \lambda_3 \]
Step 1: Compute determinant of \(A\)
Given eigenvalues: \[ 1, 2, 3 \] \[ \det(A) = 1 \times 2 \times 3 = 6 \] Concept 2: Determinant of powers
For any square matrix: \[ \det(A^k) = (\det A)^k \]
Step 2: Apply the formula for \(A^2\)
\[ \det(A^2) = (\det A)^2 = 6^2 = 36 \]
Step 3: Alternative understanding
Eigenvalues of \(A^2\) are: \[ 1^2, 2^2, 3^2 = 1, 4, 9 \] \[ \det(A^2) = 1 \times 4 \times 9 = 36 \] Conclusion: \[ \det(A^2) = 36 \]