Question

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 \(C^{2} .\)