Exponential of a specific matrix

Question

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 $t$

please help :)

Answer 1

Hint

• 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.

Answer 2

You can use the Laplace transform; $$\exp(At)=\mathscr{L}^{-1}\{ (sI-A)^{-1} \}.$$