Suppose the collection $\left\{A_{1}, \cdots, A_{k}\right\}$ forms a group under matrix multiplication, where each $A_{i}$ is an $n \times n$ real matrix. Let $\displaystyle{}A=\sum_{i=1}^{k} A_{i}$.
- Show that $A^{2}=k A$.
- If the trace of $A$ is zero, then show that $A$ is the zero matrix.