(I apologize in advance for the formatting. This is my first post here. Advice is welcome in that regard.)
This is the second part of a two-part question. The first part asks the following:
"If 2i is an eigenvalue of a real 2 x 2 matrix A, find A^2."
For this part, I simply used the fact that the eigenvalues for a matrix {{a,c},{d,a}} are given by a +/- sqrt(cd). I then let a = 0, c = 1, and d = 4/c = 1, giving me the matrix {{0,1},{4,0}}. Sure enough, that has the eigenvalue 2i. I, then, just multiplied it by itself to get A^2. I assume, however, that I was supposed to use the "A^2" part of the problem as some sort of a "hint" for its solution, and that's why I'm getting caught up on the second part of the problem. Anyway, the "nonzero" part is the issue for me because the matrix I used for the first part clearly has zero entries, and I can't figure out how one is supposed to construct a matrix with the given properties without zeroes in the "a" positions (seeing as the eigenvalues are given by a +/- sqrt(cd)).
So, that's my problem.