Horn and Johnson in "Matrix Analysis" leave as a exercise this proof:
Let A be a real matrix with characteristic polynomial $p(\lambda)$ where $$p(\lambda) = \lambda^n + c_{1}\lambda^{n-1} + c_2\lambda^{n-2}+\cdots+c_n.$$ Let $E_k = \sum C_{kk}$ where $C_{kk}$ means a $k$-by-$k$ principal minor of $A$, and the summation is over all $k$-by-$k$ principal minors. Then, $$p(\lambda) = \lambda^n + E_1\lambda^{n-1}+E_2\lambda^{n-2}+\cdots+E_n.$$
The authors say that this can be proved by mathematical induction, using the Laplace expansion.
I have written out the base case and the induction hypothesis. My assumption is that the induction is on the dimension of the matrix, although now I am not sure at this point as I don't know what to do from here.
Could anyone give me a hint or help as to what I should do next?