Question

Let T: R^3 to R^3 be the linear transformation defined by

T(x,y,z)= (-x,x-y, 3x+2y+z)

Check whether T satifies the polynomial (x-1)(x+1)^2. Also the find of minimal polynomial of T.

Answer 1

$T=x\begin{pmatrix} -1 \\ 1\\ 3 \end{pmatrix}+y\begin{pmatrix} 0 \\ -1\\ 2 \end{pmatrix}+z\begin{pmatrix} 0 \\ 0\\ 1 \end{pmatrix}$

$T=\begin{pmatrix} -1&0 & 0 \\ 1&-1 & 0\\ 3&2&1 \end{pmatrix}$

$T-1=\begin{pmatrix} -2&0& 0 \\ 1&-2 & 0\\ 3&2&0 \end{pmatrix}$

$(T+1)^2=\begin{pmatrix} 0&0&0 \\ 1&0&0\\ 3&2&2 \end{pmatrix}^2=\begin{pmatrix} 0&0&0\\ 0&0& 0\\ 8&4&4 \end{pmatrix}$

$(T-1)(T+1)^2=\begin{pmatrix} -2&0& 0 \\ 1&-2 & 0\\ 3&2&0 \end{pmatrix}\begin{pmatrix} 0&0&0 \\ 0&0 & 0\\ 8&4&4 \end{pmatrix}=\begin{pmatrix} 0&0& 0 \\ 0&0& 0\\ 0&0&0 \end{pmatrix}$

T satisfies the polynomial $(x-1)(x+1)^2$

$(T-1)(T+1)=\begin{pmatrix} -2&0&0 \\ 1&-2 & 0\\ 3&2&0 \end{pmatrix}\begin{pmatrix} 0&0& 0\\ 1&0&0 \\ 3&2&2 \end{pmatrix}=\begin{pmatrix} 0&0 & 0\\ -2&0&0\\ 2&0&0 \end{pmatrix}\ne\begin{pmatrix} 0&0 & 0 \\ 0&0& 0\\ 0&0&0 \end{pmatrix}$

$\therefore$ Minimal polynomial is $(x-1)(x+1)^2$