Question

Let \( M \) be a \( 2 \times 2 \) matrix. Its trace is 6 and its determinant has value 8. Its eigenvalues are:

• A. 2 and 4
• B. 3 and 3
• C. 2 and 6
• D. -2 and 3

Hint:

For a 2×2 matrix, the trace equals the sum and the determinant equals the product of eigenvalues.

Answer 1

The Correct Option is A

Step 1: Use properties of eigenvalues.
For a \( 2 \times 2 \) matrix with eigenvalues \( \lambda_1 \) and \( \lambda_2 \): \[ \lambda_1 + \lambda_2 = \text{trace}(M) = 6, \] \[ \lambda_1 \lambda_2 = \text{det}(M) = 8. \]
Step 2: Solve for eigenvalues.
Let \( \lambda_1 = 2, \lambda_2 = 4 \). Then: \[ \lambda_1 + \lambda_2 = 2 + 4 = 6, \quad \lambda_1 \lambda_2 = 8. \] Hence, both conditions are satisfied.
Step 3: Final Answer.
Eigenvalues of \( M \) are 2 and 4.