Another challenge problem, inspired by an earlier question.

Given arbitrary 2x2 matrices $\displaystyle A$ and $\displaystyle B$, prove that

$\displaystyle AB+BA=\beta A+\alpha B+(\gamma-\alpha\beta)I$

where $\displaystyle \alpha$ is the trace of $\displaystyle A$, $\displaystyle \beta$ is the trace of $\displaystyle B$ and $\displaystyle \gamma$ is the trace of $\displaystyle AB$ or of $\displaystyle BA$. $\displaystyle I$ is of course the 2x2 identity matrix.

Although a brute force approach would work, a more elegant solution would be welcome.

Enjoy.

Moderator approved. CB