I am trying to show that in the characteristic polynomial of $\det (A-\lambda I)$ the coefficient of $\lambda^{n-1}$ is $(-1)^{n-1} \operatorname{Tr} A$.
I'm given the hint to use the Leibniz formula for this determinant, $$\sum_{\sigma \in S_n} \operatorname{sgn}(\sigma) \prod_{i=1}^n (a_{i\sigma(i)}-\lambda \delta_{i\sigma(i)}).$$
I don't really even know where to start. I have little experience with the Leibniz formula as I would usually compute determinants using Laplace expansion.