For the matrix $A = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}$, the trace of the matrix $A^3$ is ________ (answer in integer).
Answer Key:
35
Step-by-Step Solution & Explanation
Matrix $A$ is upper triangular, so its eigenvalues are the diagonal elements:
$\lambda_1 = 2, \quad \lambda_2 = 3$
By the spectral mapping theorem, the eigenvalues of $A^3$ are:
$\lambda_1^3 = 2^3 = 8, \quad \lambda_2^3 = 3^3 = 27$
The trace of a matrix is the sum of its eigenvalues:
$\operatorname{trace}(A^3) = \lambda_1^3 + \lambda_2^3 = 8 + 27 = 35$