I did a problem in class today that evaluated [tex]f(t)=e^{At}[/tex] for [tex]A_{2,2}=\begin{bmatrix}2&1 \\-1&4 \end{bmatrix}[/tex] to a matrix form.(adsbygoogle = window.adsbygoogle || []).push({});

The answer I got was:

[tex]f(t)=\begin{bmatrix}e^{3t}-te^{3t}&te^{3t} \\-te^{3t}&e^{3t}+te^{3t} \end{bmatrix}[/tex]

Factoring we have:

[tex]f(t)=e^{3t}\begin{bmatrix}1-t&t \\-t&1+t \end{bmatrix}[/tex]

My question is, is there some simple general expression for simplifying [tex]e^{At}[/tex] to a matrix form? Maybe something that resembles [tex]e^{tA_{2,2}}=e^{\lambda t}\begin{bmatrix}1-t&t \\-t&1+t \end{bmatrix}[/tex]

but for any size matrix.

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# Evaluated e^At

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