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 $A$?
Best Answer
Use the fact that $\begin{vmatrix} a & b+e \\ c & d+f \end{vmatrix} = \begin{vmatrix} a & b \\ c & d \end{vmatrix} + \begin{vmatrix} a & e \\ c & f \end{vmatrix} $
We can use this fact to separate out powers of $\lambda$. Following is an example for $2 \times 2$ matrix. $$ \begin{vmatrix} a-\lambda & b \\ c & d-\lambda \end{vmatrix} = \begin{vmatrix} a & b \\ c & d-\lambda \end{vmatrix} + \begin{vmatrix} -\lambda & b \\ 0 & d-\lambda \end{vmatrix} = \begin{vmatrix} a & b \\ c & d \end{vmatrix} + %% \begin{vmatrix} a & 0 \\ c & -\lambda \end{vmatrix} + %% \begin{vmatrix} -\lambda & b \\ 0 & d \end{vmatrix} + \begin{vmatrix} -\lambda & 0 \\ 0 & -\lambda \end{vmatrix} $$
This decompose $det$ expression into sum of various powers of $\lambda$.
Now try it with a $3 \times 3$ matrix and then generalize it.