Define the exponential of a matrix A to be $\displaystyle e^{A} = I + A + \frac{1}{2!}A^{2}+\frac{1}{3!}A^{3}+...$.

Let A be the following matrix $\displaystyle \left(\begin{array}{cc}3 & -2 \\ -2 & 3\end{array}\right)$. Show that the eigen values of A are $\displaystyle \lambda_{1} = 1 $ and $\displaystyle \lambda_{2} = 5$ with associated eigenvectors $\displaystyle \left(\begin{array}{c}1\\1\end{array}\right),\left (\begin{array}{c}1\\-1\end{array}\right)$. Hence or otherwise calculate $\displaystyle e^{A}$.

Use previous to solve the system of differential equations (with respect to t)

$\displaystyle y'_{1}=3y_{1} + -2y_{2}$

$\displaystyle y'_{2}=-2y_{1}+3y_{2} $

with the initial conditions $\displaystyle y_{1}(0)=2$, and $\displaystyle y_{2}(0)=0$, by writing the system as

$\displaystyle Y'=AY, Y(0)=Y_{0}$

for the matrix A above, and $\displaystyle Y_{0} = \left(\begin{array}{c}2\\0\end{array}\right)$ and noting that a solution is $\displaystyle Y(t)=e^{tA}Y_{0}$

I can do the eigenvalues and eigenvectors part, and can calculate $\displaystyle e^{A}$ since $\displaystyle e^{A}=Xe^{D}X^{-1}$,

but I don't get the differential equations part...

Any help would be much appreciated...