eigenvalues of a matrix with zero $k^{th}$ power For a matrix $A$, where $A^k=0$, $k\ge1$, need prove that $trace(A)=0$; i.e sum of eigenvalues is zero. How do you approach this problem?

Do you know that the trace of a matrix is the sum of it's eigenvalues counted with multiplicity?  Think about the Jordan Canonical Form of your matrix to see why this is so.
Now for a matrix that satisfies $A^k = 0$ prove that $0$ is its only eigenvalue.
