Question

Find a matrix A whose minimal polynomial is t⁴-5t³-2t²+7t+4

Answer 1

Here the minimal polynomial is $t^4-5t^3-2t^2+7t+4$ .

Equating it to 0, and finding the roots we get:-

$$t^4-5t^3-2t^2+7t+4=0\\ \Rightarrow t=1.4\>or\>t=5.093$$

Now, by forceful calculation we get that that the minimal polynomial can be factored as:-

$(t-1.4)^3(t-5.093)$

So, the values in the primary diagonal will be:-

$1.4,1.4,1.4,5.093$

So, our required matrix A is:-

$A=\begin{bmatrix} 1.4 & 0 & 0 & 0\\ 0 & 1.4 & 0 & 0\\ 0 & 0 & 1.4 & 0\\ 0 & 0 & 0 & 5.093 \end{bmatrix}$