Compute $e^A$ where $A$ is a given $2\times 2$ matrix Compute $e^A$ where $A=\begin{pmatrix} 1 &0\\ 5 & 1\end{pmatrix}$
definition
Let $A$ be an $n\times n$ matrix. Then for $t\in \mathbb R$,
$$e^{At}=\sum_{k=0}^\infty \frac{A^kt^k}{k!}\tag{1}$$

Another way of proving this...
we know from  the property of $e$, that if $S,T$ are linear operators and $ST=TS$ then 
$$e^{S+T}=e^S.e^T$$
Letting $S=I$ and $T=\begin{pmatrix}0&0\\5&0\end{pmatrix}$
Then $$e^{S+T}=e^S.e^T=e^I.e^{\begin{pmatrix}0&0\\5&0\end{pmatrix}}=e(I+T)=e\begin{pmatrix}1&0\\5&1\end{pmatrix}$$
Since, $T^2=0=T^3=\ldots$
