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Let $C$ be a $2 \times 2$ matrix of real numbers. Give a proof or counterexample to each of the following assertions:
a) $\operatorname{det}\left(C^{2}\right)$ is non-negative.
b) trace $\left(C^{2}\right)$ is non-negative.
c) All of the elements of $C^{2}$ are non-negative.
d) All the eigenvalues of $C^{2}$ are non-negative.
e) If $C$ has two distinct eigenvalues, then so does $C^{2} .$
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