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\begin{document}
Homework problem from Chapter 5

\bigskip

Let $A$ be a square matrix with real entries. We can define a matrix $\exp A$
by the \textquotedblleft power series\textquotedblright%
\[
\exp A=I+A+\frac{1}{2}A^{2}+\frac{1}{3!}A^{3}+\cdots
\]
Suppose $A$ is diagonalizable with eigenvalues $\lambda_{1},\ldots,\lambda
_{n}$ (not necessarily distinct). 

a. Give a general formula for $A^{n}$ of the form $Q^{-1}D_{n}Q$ where $D_{n}$
is diagonal. Be sure to specify what $Q$ and $D_{n}$ are precisely.

b. Give a general formula for $\exp A$ of the form $Q^{-1}DQ$ where $D$ is diagonal.

c. Calculate $\exp\left(
\begin{array}
[c]{cc}%
3 & 12\\
0 & -1
\end{array}
\right)  $.


\end{document}