Let $A, B, C$ be $2 \times 2$ matrices, real or complex. Define the trace $\operatorname{tr} C$ to be the sum of diagonal entries $C_{11}+C_{22}$. Define the commutator $[A, B]$ to be the difference $A B-B A$. Give the definition of the eigenvalues of a $2 \times 2$ matrix and prove that it can have at most two distinct eigenvalues. Prove that a) $\operatorname{tr}[A, B]=0$, b) $\operatorname{tr} C$ equals the sum of the eigenvalues of $C$, c) if all eigenvalues of $C$ are equal to 0 then $C^{2}=0$, d) either $[A, B]$ is a diagonalisable matrix or the square $[A, B]^{2}=0$, e) $[A, B]^{2}=\alpha I$ where $\alpha \in \mathbb{C}$ and $I$ is the unit matrix.