Question

find the minimal polynomial of the linear operator t : R³ "- R³" define by t (x,y,z) =(x+2y+3z, 4y+5z,6 z).is t

Answer 1

$\begin{pmatrix} X+2y+3z \\ 4y+5z\\ 6z \end{pmatrix}$ $=x\begin{pmatrix} 1\\ 0\\ 0 \end{pmatrix}+y\begin{pmatrix} 2\\ 4 \\ 0 \end{pmatrix}+z \begin{pmatrix} 3 \\ 5 \\ 6 \end{pmatrix}$

Transforming matrix

$A=\begin{pmatrix} 1&2&3 \\ 0&4&5\\ 0&0&6 \end{pmatrix}$

Characteristic equation of A is

$\>\>\begin{vmatrix} A-I\lambda \\ \end{vmatrix}=\begin{vmatrix} 1-\lambda&&2&&3 \\ 0&&4-\lambda&&5 \\ 0&&0&&6-\lambda \end{vmatrix}=0$

Expanding

$(1-\lambda)[(4-\lambda)(6-\lambda)-0]+2(0)+3(0)=0$

$\implies(1-\lambda)(4-\lambda)(6-\lambda)=0$

The distincts roots of A are also roots of the minimum polynomial $g(x)$

$\therefore\>g(x)=(x-1)(x-4)(x-6)$