The trace is the sum of the diagonal elements, so (when $z$ is real) it's always true for at least one $i$, and the only way it can be true for all of them is that all diagonal elements are equal. For example, this is the case for the matrix $$ A = \pmatrix{0 & 0 & 1 & -1\cr 0 & 0 & 1 & -1\cr 1 & 1 & 0 & 0\cr -1 & -1 & 0 & 0\cr} $$$$ A = \pmatrix{3 & 0 & 1 & -1\cr 0 & 3 & 1 & -1\cr 1 & 1 & 3 & 0\cr -1 & -1 & 0 & 3\cr} $$
