Es sei ${\displaystyle {}M=(a_{ij})_{ij}}$, somit ist

${\displaystyle {}sM={\begin{pmatrix}sa_{11}&sa_{12}&\ldots &sa_{1n}\\sa_{21}&sa_{22}&\ldots &sa_{2n}\\\vdots &\vdots &\ddots &\vdots \\sa_{n1}&sa_{n2}&\ldots &sa_{nn}\end{pmatrix}}\,.}$

Sei zunächst

${\displaystyle {}s\neq 0\,.}$

Es ist

${\displaystyle {}{\begin{aligned}\chi _{sM}&=\det(XE_{n}-sM)\\&=\det {\begin{pmatrix}X-sa_{11}&sa_{12}&\ldots &sa_{1n}\\sa_{21}&X-sa_{22}&\ldots &sa_{2n}\\\vdots &\vdots &\vdots &\vdots \\sa_{n1}&sa_{n2}&\ldots &X-sa_{nn}\end{pmatrix}}\\&=\det {\begin{pmatrix}s{\left({\frac {X}{s}}-a_{11}\right)}&sa_{12}&\ldots &sa_{1n}\\sa_{21}&s{\left({\frac {X}{s}}-a_{22}\right)}&\ldots &sa_{2n}\\\vdots &\vdots &\vdots &\vdots \\sa_{n1}&sa_{n2}&\ldots &s{\left({\frac {X}{s}}-a_{nn}\right)}\end{pmatrix}}\\&=s^{n}\cdot \det {\begin{pmatrix}{\frac {X}{s}}-a_{11}&a_{12}&\ldots &a_{1n}\\a_{21}&{\frac {X}{s}}-a_{22}&\ldots &a_{2n}\\\vdots &\vdots &\vdots &\vdots \\a_{n1}&a_{n2}&\ldots &{\frac {X}{s}}-a_{nn}\end{pmatrix}}\\&=s^{n}\cdot \chi _{M}{\left({\frac {X}{s}}\right)}.\end{aligned}}}$

Hier steht also das charakteristische Polynom zu ${\displaystyle {}M}$, wobei man die Variable überall durch ${\displaystyle {}{\frac {X}{s}}}$ ersetzt, und das Ganze mit ${\displaystyle {}s^{n}}$ multipliziert. Daher ist

${\displaystyle {}{\begin{aligned}\chi _{sM}&=s^{n}{\left({\left({\frac {X}{s}}\right)}^{n}+c_{n-1}{\left({\frac {X}{s}}\right)}^{n-1}+c_{n-2}{\left({\frac {X}{s}}\right)}^{n-2}+\cdots +c_{2}{\left({\frac {X}{s}}\right)}^{2}+c_{1}{\left({\frac {X}{s}}\right)}+c_{0}\right)}\\&=X^{n}+sc_{n-1}X^{n-1}+s^{2}c_{n-2}X^{n-2}+\cdots +s^{n-2}c_{2}X^{2}+s^{n-1}c_{1}X+s^{n}c_{0}.\end{aligned}}}$

Dieser Zusammenhang gilt auch bei ${\displaystyle {}s=0}$,

${\displaystyle {}sM}$

${\displaystyle {}X^{n}}$