Question

The matrix \( M \) is defined as \[ M = \begin{bmatrix} 1 & 3 \\ 4 & 2 \end{bmatrix} \] and has eigenvalues 5 and -2. The matrix \( Q \) is formed as \[ Q = M^3 - 4M^2 - 2M \] Which of the following is/are the eigenvalue(s) of matrix \( Q \)? \\

• A. 15
• B. 25
• C. -20
• D. -30

Hint:

When calculating eigenvalues for a matrix function, substitute the eigenvalues of the original matrix into the expression for the matrix function.

Answer 1

The Correct Option is A, C

We are given that the matrix \( M \) has eigenvalues 5 and -2. We need to calculate the eigenvalues of matrix \( Q \), which is defined as: \[ Q = M^3 - 4M^2 - 2M. \] Step 1: Use the properties of eigenvalues. If \( \lambda \) is an eigenvalue of matrix \( M \), then \( \lambda^n \) is an eigenvalue of \( M^n \). Thus, we can compute the eigenvalues of \( Q \) by substituting the eigenvalues of \( M \) into the expression for \( Q \). Given that the eigenvalues of \( M \) are 5 and -2, we calculate the corresponding eigenvalues for \( Q \) as follows: For \( \lambda = 5 \): \[ Q_{\text{eigenvalue}} = 5^3 - 4(5^2) - 2(5) = 125 - 100 - 10 = 15. \] For \( \lambda = -2 \): \[ Q_{\text{eigenvalue}} = (-2)^3 - 4(-2)^2 - 2(-2) = -8 - 16 + 4 = -20. \] Thus, the eigenvalues of \( Q \) are 15 and -20. Final Answer: (A) 15
(C) -20