Again, the wording to me is odd. Are these matrices all square matrices i.e.
n×n? I presume from the fact that you talk about "3 nonzero eigenvalues" that you are considering
3×3 matrices here.
Assuming that these matrices are 3x3: I agree with your thoughts on (a). As for (d), it's true that
S−1AS exists and is diagonal; therefore
A is diagonalisable, but I'd prefer to say something along the lines of "there are 3 linearly independent eigenvectors of A (namely, columns of S) and these must form a basis of 3 dimensional space, so A is diagonalisable" as this is often quotable whereas your answer requires this condition to hold.
For (b), why must A have 3 non-zero eigenvalues? Can you come up with an example of a 3x3 matrix with an eigenvalue of zero that also has 3 L.I. eigenvectors?
And for what you've written about (c), what if some of those non-zero eigenvalues were repeated? Can you come up with a 3x3 matrix with degenerate eigenvalues that do not produce a basis of eigenvectors for 3D space? This doesn't give you an answer but does give you a cause for concern about the diagonalisability of S.