Characteristic polynomial of a matrix is monic? Given a $n \times n$ matrix A, I need to show that its characteristic polynomial, defined as $P_A (x) = det (xI-A)$ is monic. I am trying induction. But no clue after induction hypothesis.

You have 
$$c_A(\lambda) = \det(\lambda I - A)
= \sum_{\sigma\in \Sigma_n} {\rm sgn}(\sigma)
\prod_{i=1}^n
(\lambda \delta_{i\sigma(i)} - A_{i\sigma(i)}),$$
where $\Sigma_n$ is the permutation group on $n$ letters.  The only term in the sum giving a $\lambda^n$ term is the identity permutation.  Hence the polynomial is monic.
