50. Let A be a real 2x2 matrix. Which of the following statements must be true?

I. All of the entries of A^2 are nonnegative.

II. The determinant of A^2 is nonnegative.

III. If A has two distinct eigenvalues, then A^2 has two distinct eigenvalues.

A) I only B) II only C) III only D) II and III only E) I, II, III

I guessed D, but the answer is B, only II must be true.

I wasn't able to prove or disprove any of them to myself except I, which is easy to think of a counterexample for. II seemed to be true, and since I couldn't disprove III I guessed that both were true. Is there any way short of thinking up a counterexample to show that III is false? And is there a way to show that II is true? I guessed that it was, but couldn't show it.

Thanks.