How to find the eigenvalues of the following $4 \times 4$ matrix?
\begin{pmatrix} pq+1&&0&& 0&& -pq\\ 0&& pq+p&& -(p-1)&& -pq\\ 0&& -(q-1) && pq+q&& -pq\\ -(p-1)(q-1) && -(q-1) && -(p-1) && pq\\ \end{pmatrix}
The characteristic polynomial is $$\det\begin{pmatrix} x-(pq+1)&&0&& 0&& pq\\ 0&& x-(pq+p)&& (p-1)&& pq\\ 0&& (q-1) && x-(pq+q)&& pq\\ (p-1)(q-1) && (q-1) && (p-1) && x-pq\\ \end{pmatrix}$$
But I am unable to understand how to simplify the above determinant and get the roots of the above.
Can someone please help in this?
The characteristic polynomial is given by $$ f(t)=t^4 + t^3( - 4pq - p - q - 1) + t^2(5p^2q^2 + 3p^2q + 3pq^2 + 4pq + 2 p + 2q - 1) + t( - 2p^3q^3 - 2p^3q^2 - 2p^2q^3 - 5p^2q^2 - 5p^2q - 5pq^ 2 + 2pq - p - q + 1) + 2pq(p^2q^2 + p^2q + pq^2 + p + q - 1). $$ This can be factored as $$ f(t)=- (2pq - t)(pq + p + q - t - 1)(pq - t + 1)(t - 1), $$ so that the roots are obvious.