Let $A$ be an $n \times n$ matrix with characteristic polynomial $f(t)=(-1)^nt^n+b_{n-1}t^{n-1}+\cdots+b_1t+b_0$
Define $A=(a_{ij})$. Show that $f(t)=(a_{11}-t)(a_{22}-t)\cdots(a_{nn}-t)+q(t)$, where $\deg q(t)\le n-2$
I am not sure where to start this problem. What is $q(t)$ and where does it come from?
Also, how do I apply the definition of $A=(a_{ij})$?