## Companion Matrix for a Polynomial

## Problem 85

Consider a polynomial

\[p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,\]
where $a_i$ are real numbers.

Define the matrix

\[A=\begin{bmatrix}

0 & 0 & \dots & 0 &-a_0 \\

1 & 0 & \dots & 0 & -a_1 \\

0 & 1 & \dots & 0 & -a_2 \\

\vdots & & \ddots & & \vdots \\

0 & 0 & \dots & 1 & -a_{n-1}

\end{bmatrix}.\]

Then prove that the characteristic polynomial $\det(xI-A)$ of $A$ is the polynomial $p(x)$.

The matrix is called the ** companion matrix** of the polynomial $p(x)$.

Add to solve later