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 $t\in \mathbb R$,
$$e^{At}=\sum_{k=0}^\infty \frac{A^kt^k}{k!}\tag{1}$$

i will turn it into $$\frac{dx}{dt} = x, \frac{dy}{dt} = 5x + y$$ which has solution $$x = Ae^t, \frac{d}{dt}\left(ye^{-t}\right) = 5A \to y = Be^t + 5At e^t $$
we can write $$\pmatrix{x\\y} = \pmatrix{e^t&0\\5te^t&e^t}\pmatrix{x(0)\\y(0)} $$ by a property of $e^{At}$ this implies $$ e^{tA} = \pmatrix{e^t&0\\5te^t&e^t},\, e^{A} = \pmatrix{e&0\\5e&e} $$
