Computing the matrix exponential e^A

[mprm86]

Calculate $e^A$ if $A = \left(\begin{array}{cc}0 & -2\\1 & 3\end{array}\right)$

Maybe using that $$e^x = \sum_{n=0}^\infty \frac{x^n}{n!}$$, then

$$e^A = \sum_{n=0}^\infty \frac{\left(\begin{array}{cc}0 & -2\\1 & 3\end{array}\right)^n}{n!}$$
but i don't know if this is the right way for doing this. Please help me. Thanks.

 

[AKG]

That's precisely it.

 

[mprm86]

Ok, thanks.

 

[HallsofIvy]

Although you may have difficulty calculating $$\left(\begin{array}{cc}0 & -2\\1 & 3\end{array}\right)^n$$
Knowing that its eigenvalues are 1 and 2 (with eigenvectors <2, -1> and <1, -1> respectively) will help.