Compute $e^{Ax}$ for the $2 \times 2$ matrix $A$ Consider the real $2 \times 2$ matrix $A$  $$A=\begin{pmatrix} 1 & 0 \\ 0 & -1\end{pmatrix}$$
I want to compute $e^{Ax}$. 
I can see that $A^2=I$ which looks like it should be useful.
Using the definition of the matrix exponential $$e^{Ax}=I+Ax+\frac{(Ax)^2}{2!}+ \cdots$$ I find that $$e^{Ax}=I\left(1+\frac{x^2}{2!}+ \cdots\right)+A\left(x+\frac{x^3}{3!}+ \cdots\right)$$
Is there anything I can do from here?

We have $$e^{Ax}=\begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix}\left(1+\frac{x^2}{2!}+ \cdots\right)+\begin{pmatrix} 1 & 0 \\ 0 & -1\end{pmatrix}\left(x+\frac{x^3}{3!}+ \cdots\right)$$ This can be written as $$\begin{pmatrix} 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+... & 0 \\ 0 & 1+(-x)+\frac{x^2}{2!}+\frac{(-x)^3}{3!}+...\end{pmatrix}$$ so $$\boxed{e^{Ax}=\begin{pmatrix} e^x & 0 \\ 0 & e^{-x}\end{pmatrix}}$$
