Given a matrix A, I need determine the number of positive, negative and zero eigen value of it without actually computing it?. The matrix is given below
\begin{pmatrix}0&...&0 & 1\\ 0&...&0 & 2\\ .&.&. &.\\ .&.&. &.\\ 0&...&0 &n-1\\ 1&...&n-1&n \end{pmatrix}