Exercise 33

We define the trace of a $2 \times 2$ matrix to be the sum of its diagonal entries. That is, the trace of $$A =\begin{pmatrix} a & b \\ c & d \end{pmatrix}$$ is $\trace(A) = a + d$. Show that $\trace(AB) = \trace(BA)$ for any $2 \times 2$ matrices $A$ and $B$.