Question

If the eigenvalues of a \(2 \times 2\) matrix \(P\) are 4 and 2, then the eigenvalues of the matrix \(P^{-1}\) are

• A. 0, 0
• B. 0.0625, 0.25
• C. 0.25, 0.5
• D. 2, 4

Hint:

For a matrix \( P \), if its eigenvalues are \( \lambda_1 \) and \( \lambda_2 \), the eigenvalues of \( P^{-1} \) are \( \frac{1}{\lambda_1} \) and \( \frac{1}{\lambda_2} \).

Answer 1

The Correct Option is C

For a square matrix \(P\), if its eigenvalues are \( \lambda_1 \) and \( \lambda_2 \), the eigenvalues of its inverse matrix \( P^{-1} \) are given by the reciprocals of the eigenvalues of \(P\). That is, the eigenvalues of \( P^{-1} \) are \( \frac{1}{\lambda_1} \) and \( \frac{1}{\lambda_2} \). Given that the eigenvalues of \( P \) are 4 and 2, the eigenvalues of \( P^{-1} \) are: \[ \frac{1}{4} = 0.25 \quad \text{and} \quad \frac{1}{2} = 0.5. \] Thus, the correct answer is (C).