Question

Find a matrix A whose minimal polynomial is t^3-5t^2+6t+8

Answer 1

The matrix associated to a generic minimal polynomial $p(x)=a_0+a_1x+a_2x^2+\cdots +a_{n-1}x^{n-1}+a_nx^n$ is given as;

$A=\begin{pmatrix} 0&-a_0 \\ I&-a \end{pmatrix}$

Where $I$ is the $(n-1) \times (n-1)$ identity matrix and $a=(a_1,\cdots, a_{n-1})^T$. A is an $n\times n$ matrix.

So, for the question,

$a_0=8,a=(6,-5)^T, I=\begin{pmatrix} 1&0\\ 0&1 \end{pmatrix}$

$A=\begin{pmatrix} 0&0&-8\\ 1&0&-6\\ 0&1&5\\ \end{pmatrix}$