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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 :)

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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.

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  • $\begingroup$ Actually, it is sufficient to diagonalize just $\mathbf A$, since for a scalar $t$, $t \mathbf A=t \mathbf P \mathbf D \mathbf {P^{-1}}=\mathbf P t \mathbf D \mathbf {P^{-1}}$ $\endgroup$ – G Cab Nov 1 '16 at 11:18
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You can use the Laplace transform; $$\exp(At)=\mathscr{L}^{-1}\{ (sI-A)^{-1} \}.$$

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