I've been looking around stack exchange, but I've been having some issues with this question.
Let $A\in M_{n,n}(K)$, for some field $K$. We write $$c_A = (-1)^n \left(t^n + \sum_{i=0}^{n-1} c_i t^i \right)$$ with $c_i \in K$.
To show $c_{n-1} = -\text{tr}(A)$.
I'm not quite sure how to go about this. My idea has been using induction. For $n=2$ the base case, we have that the characteristic polynomail is $c_A(t) = t^2 - \text{tr}(A) t + \det(A)$. Here, the coefficient of $t$, or $c_{n-1} = -\text{tr}(A)$. So in this case it works. (I think?)
The inductive step here would be assuming that it holds for some $A\in M_{n+1, n+1 }(K)$. Here, the characteristic polynomail is $$c_A(t) = (a_{11} - t) \det\left(\begin{bmatrix} a_{22} - t & \ldots & a_{2 \, n} & a_{2 \; (n + 1)} \\ a_{32} & a_{33} - t & \ldots & a_{3 \; (n + 1)} \\ \ldots \\ a_{(n + 1) \; 2} & \ldots & a_{(n + 1) \, n} & a_{(n + 1) \; (n + 1)} - t\end{bmatrix}\right).$$
Is this correct? How would I continue from here? I am quite stuck.