## Diagonalize a 2 by 2 Matrix $A$ and Calculate the Power $A^{100}$

## Problem 466

Let

\[A=\begin{bmatrix}

1 & 2\\

4& 3

\end{bmatrix}.\]

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

**(b)** Find eigenvectors for each eigenvalue of $A$.

**(c)** Diagonalize the matrix $A$. That is, find an invertible matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.

**(d)** Diagonalize the matrix $A^3-5A^2+3A+I$, where $I$ is the $2\times 2$ identity matrix.

**(e)** Calculate $A^{100}$. (You do not have to compute $5^{100}$.)

**(f)** Calculate

\[(A^3-5A^2+3A+I)^{100}.\]
Let $w=2^{100}$. Express the solution in terms of $w$.