For a given $n \times n$-matrix $A$, and $J\subseteq\{1,...,n\}$ let us denote by $A[J]$ its principal minor formed by the columns and rows with indices from $J$.
If the characteristic polynomial of $A$ is $x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$, then why $a_k=(-1)^{n-k}\sum_{|J|=n-k}A[J],$ that is, why is each coefficient the sum of the appropriately sized principal minors of $A$?