Characteristic polynomial/Companion matrix/Exercise

Show that the characteristic polynomial of the so-called companion matrix

${}M={\begin{pmatrix}0&1&0&\ldots &0\\0&0&1&\ldots &0\\\vdots &\vdots &\ddots &\ddots &\vdots \\0&0&\ldots &0&1\\-a_{0}&-a_{1}&\ldots &-a_{n-2}&-a_{n-1}\end{pmatrix}}\,$

equals

${}\chi _{M}=X^{n}+a_{n-1}X^{n-1}+\cdots +a_{1}X+a_{0}\,.$

Create a solution