## Diagonalize the Upper Triangular Matrix and Find the Power of the Matrix

## Problem 583

Consider the $2\times 2$ complex matrix

\[A=\begin{bmatrix}

a & b-a\\

0& b

\end{bmatrix}.\]

**(a)** Find the eigenvalues of $A$.

**(b)** For each eigenvalue of $A$, determine the eigenvectors.

**(c)** Diagonalize the matrix $A$.

**(d)** Using the result of the diagonalization, compute and simplify $A^k$ for each positive integer $k$.