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 property of eigenvalues for a polynomial of a matrix. If \(\lambda\) is an eigenvalue of a matrix \(A\), then for any polynomial \(P(x)\), the corresponding eigenvalue of the matrix \(P(A)\) is \(P(\lambda)\).
Step 2: Define the polynomial and list the eigenvalues of A. The matrix polynomial is \(P(A) = 3I - 2A + A^2\). The corresponding scalar polynomial is \(P(\lambda) = 3 - 2\lambda + \lambda^2\). 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. These correspond to statements A, B, and D. Statement C (8) is not an eigenvalue.