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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.
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