I am trying to understand lemma 1.9 from these notes https://ocw.mit.edu/courses/mathematics/18-314-combinatorial-analysis-fall-2014/readings/MIT18_314F14_mt.pdf
I understand the proof up to the step where it says "Factor out $−x$ from the last row, yielding a matrix $N(x)$ satisfying det$(M−xI)$ = $−x$det$(N(x))$. Hence the coefficient of $x$ in det$(M − xI)$ is given by $−$det$(N(0))$."
I am unfamiliar with the notation $N(x)$ despite attempts to find an explanation. Could someone please explain what $N(x)$ means and how this describes the coefficient of x.
Thank you in advance.
Replace the last row of $M-xI$ by a row of all $1$s. You'll end up with another matrix whose entries are either constant or linear in $x$. We call this matrix $N(x)$ to emphasize its dependence on $x$.
Since the determinant is multilinear and alternating, the construction of $N(x)$ given in the proof (add all the rows to the last row and factor out a $-x$ from the resulting last row) shows that $\det(M-xI) = -x\det(N(x)).$
We can also show that $\det(N(x))$ is a polynomial in $x$ (depending on how you've defined the determinant, this might be trivial). The constant term of the polynomial $\det(N(x))$ is $\det(N(0))$ and therefore the coefficient of $x$ in $-x\det(N(x))$ is $-\det(N(0)).$