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

Step 1: Understanding the Question
The question provides the eigenvalues of a matrix \(A\) and asks for the eigenvalues of the matrix \(A^2\).
Step 2: Key Formula or Approach
There is a fundamental property of eigenvalues: If \( \lambda \) is an eigenvalue of a matrix \(A\), then \( \lambda^k \) is an eigenvalue of the matrix \(A^k\) for any positive integer \(k\).
For this question, \(k=2\). So, if \( \lambda \) is an eigenvalue of \(A\), then \( \lambda^2 \) is an eigenvalue of \(A^2\).
Step 3: Detailed Explanation
The given eigenvalues of matrix \(A\) are \(\lambda_1 = 2\) and \(\lambda_2 = 3\).
To find the eigenvalues of \(A^2\), we need to square each eigenvalue of \(A\).
The first eigenvalue of \(A^2\) is:
\[ (\lambda_1)^2 = 2^2 = 4 \] The second eigenvalue of \(A^2\) is:
\[ (\lambda_2)^2 = 3^2 = 9 \] Step 4: Final Answer
The eigenvalues of the matrix \(A^2\) are 4 and 9.