Question

The eigen values of matrix \(A\) are 1, -2, 3. The eigen values of \(3I - 2A + A^2\) are:
A. 2
B. 6
C. 8
D. 11
Choose the correct answer from the options given below:

• A. A, B and D only
• B. A, B and C only
• C. A, B, C and D
• D. B, C and D only

Hint:

This property is very powerful. It means you don't need to find the matrix \(A\) itself. You can find the eigenvalues of any function of \(A\) (like \(A^2\), \(A^{-1}\), or \(e^A\)) directly from the eigenvalues of \(A\).

Answer 1

The Correct Option is A

Step 1: Recall the eigenvalue property for matrix polynomials. If \(\lambda\) is an eigenvalue of matrix \(A\), then \(P(\lambda)\) is the corresponding eigenvalue of the matrix polynomial \(P(A)\).

Step 2: Define the polynomial and list the eigenvalues of \(A\). The matrix polynomial is \(P(A) = 3I - 2A + A^2\), leading to the scalar polynomial \(P(\lambda) = 3 - 2\lambda + \lambda^2\). The eigenvalues of \(A\) are \(\lambda_1 = 1\), \(\lambda_2 = -2\), and \(\lambda_3 = 3\).

Step 3: Calculate the new eigenvalues by applying the polynomial to each eigenvalue of \(A\).

• For \(\lambda_1 = 1\): \(P(1) = 3 - 2(1) + 1^2 = 3 - 2 + 1 = 2\).
• For \(\lambda_2 = -2\): \(P(-2) = 3 - 2(-2) + (-2)^2 = 3 + 4 + 4 = 11\).
• For \(\lambda_3 = 3\): \(P(3) = 3 - 2(3) + 3^2 = 3 - 6 + 9 = 6\).

Step 4: Match the calculated eigenvalues with the given statements. The eigenvalues of \(3I - 2A + A^2\) are 2, 6, and 11, corresponding to statements A, B, and D. Statement C (8) is not an eigenvalue.