Exponential of a specific matrix

The matrix is

$$A = \begin{pmatrix} 4 & -2& \\ 3 & -1&\\ \end{pmatrix}$$

And I need to caluclate $\exp(At)$, where $t$ is real.

Not sure what I need to do after I multiply everything by $t$

• Diagonalize $At$, which will give you $P$ such that $At=PDP^{-1}$.
• Remark that if $\exp(At)=P\exp(D)P^{-1}$.
• Since $D$ is diagonal, $\exp(D)$ is easy to calculate.
• Actually, it is sufficient to diagonalize just $\mathbf A$, since for a scalar $t$, $t \mathbf A=t \mathbf P \mathbf D \mathbf {P^{-1}}=\mathbf P t \mathbf D \mathbf {P^{-1}}$ – G Cab Nov 1 '16 at 11:18
You can use the Laplace transform; $$\exp(At)=\mathscr{L}^{-1}\{ (sI-A)^{-1} \}.$$