Question 705032

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