Question

If a matrix \(A\) has eigenvalues \(2\) and \(3\), what are the eigenvalues of the matrix \(A^2\)?

• A. \(2, 3\)
• B. \(4, 9\)
• C. \(5, 6\)
• D. \(8, 27\)

Hint:

If \( \lambda \) is an eigenvalue of \(A\), then for any positive integer \(k\): \[ \text{Eigenvalues of } A^k = \lambda^k \] Thus, simply raise each eigenvalue of \(A\) to the power \(k\).

Answer 1

The Correct Option is B

Concept:
If \( \lambda \) is an eigenvalue of a matrix \(A\), then the eigenvalue of \(A^k\) is \( \lambda^k \). This follows from the eigenvalue definition: \[ A\mathbf{x} = \lambda \mathbf{x} \] Multiplying both sides by \(A\): \[ A^2 \mathbf{x} = A(\lambda \mathbf{x}) = \lambda (A\mathbf{x}) = \lambda(\lambda \mathbf{x}) = \lambda^2 \mathbf{x} \] Thus, the eigenvalues of \(A^2\) are the squares of the eigenvalues of \(A\).
Step 1: Identify the eigenvalues of \(A\).
Given: \[ \lambda_1 = 2, \qquad \lambda_2 = 3 \]
Step 2: Find eigenvalues of \(A^2\).
Square each eigenvalue: \[ \lambda_1^2 = 2^2 = 4 \] \[ \lambda_2^2 = 3^2 = 9 \]
Step 3: Write the eigenvalues of \(A^2\).
\[ \text{Eigenvalues of } A^2 = 4, 9 \] \[ \therefore \text{The correct answer is } 4, 9. \]