The matrix $z I_N - A$ is a triangular matrix with diagonal
\begin{equation}
z - A_{1, 1}, \quad
z - A_{2, 2}, \quad
\ldots, \quad
z - A_{N, N}
\end{equation}
Hence, this result is a special case of
R5072: Determinant of a triangular complex matrix. $\square$
