Proof of the formula for a $2 \times 2$ matrix exponential with trace $0$

Question

I am looking for a proof that the matrix exponential of a $2 \times 2$ trace-$0$ matrix $A$ is the following.

$$ \exp(A) = \cos(\sqrt{\det A})I+\frac{\sin(\sqrt{\det A})}{\sqrt{\det A}}A $$

Answer 1

Since $A$ is $2\times2$ and $\text{Tr}A = 0$, $A$ can be written as $$\left(\begin{matrix} a & c\\ b & -a \end{matrix}\right),$$ and $A^2 = -\vert A \vert\cdot I_2$, where $\vert A\vert = \det(A)$, and $I_2$ is the $2\times 2$ identity matrix. \begin{align*} & \exp(A) = \sum_{n=0}^\infty\frac{A^n}{n!} \\ = &\sum_{k=0}^\infty\frac{A^{2k}}{(2k)!}+\sum_{k=0}^\infty\frac{A^{2k+1}}{(2k+1)!} \\ = & \sum_{k=0}^\infty(-1)^k\frac{\vert A\vert^kI_2}{(2k)!} + \sum_{k=0}^\infty(-1)^k\frac{\vert A\vert^kA}{(2k+1)!} \\ = & \cos\left(\sqrt{\det(A)}\right)I_2 + \frac{\sin\left(\sqrt{\det(A)}\right)}{\sqrt{\det(A)}}A. \end{align*}