Question 705032

SCQMEDIUM

The matrix $A$ is given by $A = \begin{pmatrix} 1 & 2 & -3 \\ 0 & 3 & 2 \\ 0 & 0 & -2 \end{pmatrix}$ . The eigenvalues of $3A^3 + 5A^2 - 6A + 2I$ , where $I$ is the identity matrix, are

(A)

4, 9, 27

(B)

1, 9, 44

(C)

1, 10, 8

(D)

4, 110, 10

Detailed Solution

The matrix is upper triangular, so the eigenvalues are $\lambda_1 = 1$ , $\lambda_2 = 3$ , and $\lambda_3 = -2$ .

For a function $f(A) = 3A^3 + 5A^2 - 6A + 2I$ , the eigenvalues are $f(\lambda_i)$ .

For $\lambda = 1$ , $f(1) = 3(1) + 5(1) - 6(1) + 2 = 4$ .

For $\lambda = 3$ , $f(3) = 3(27) + 5(9) - 6(3) + 2 = 81 + 45 - 18 + 2 = 110$ .

For $\lambda = -2$ , $f(-2) = 3(-8) + 5(4) - 6(-2) + 2 = -24 + 20 + 12 + 2 = 10$ .

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