Calculate $\exp \left[\begin{smallmatrix} 4 & 3 \\ -1 & 2 \end{smallmatrix}\right]$

Question

Given $$ A= \begin{bmatrix} 4 & 3 \\ -1 & 2 \end{bmatrix} $$ how do I calculate $e^A$?

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I know the following formula:

If $X$ is a $2 \times 2$ matrix with trace $0$, then $$e^X = \cos{\sqrt{\det(X)}} I_2 + \dfrac{\sin{\sqrt{\det(X)}}}{\sqrt{\det(X)}} X$$ Where we interpret the coeffficient of $X$ is $1$ if $\det(X)=0$.

The hint that i got in order to calculate $e^A$ is to reduce the calculation to the case with trace $0$. I dont know how to do that ( to reduce the calculation to the traceless case). Can anyone please help, I cannot proceed.

Answer 1

One result that will get you the final part of the way it's the following:

If $M$ and $N$ are square matrices that commute, then$$e^{M+N}=e^Me^N$$

And since any matrix commutes with multiples of the identity, can you think of a way to write $A$ as the sum of a traceless matrix and a multiple of the identity matrix?