If $\displaystyle A$ is a square matrix of order $\displaystyle n$, then prove that

$\displaystyle det(\lambda I-A)=\lambda ^n-tr(A)\lambda ^{n-1}+P_{n-2}(\lambda)$

with $\displaystyle P_{n-2}$ is the polynomial of order (teh greatest) $\displaystyle n-2\text{ in }\lambda$