# jordan form question

• Oct 1st 2011, 03:04 AM
transgalactic
jordan form question
2)A)
find the definative polinomial and the minimal polinomial of
$\displaystyle A=\left(\begin{array}{ccc}3 & 1 & -1\\2 & 2 & 0\\1 & -1 & 3\end{array}\right)$
find the primary decomposition of $\displaystyle R^{3}$ in this case,and write the diagonal block matrices
similar to A
?
B)
find the jordan form of A defined in part A
and find the transformation matrices to the basis for which it represented in jordan form.
how i tried:
regarding A:
how i tried:
|tI-A|=$\displaystyle \left|\begin{array}{ccc}t-3 & -1 & +1\\-2 & t-2 & 0\\-1 & 1 & t-3\end{array}\right|$=$\displaystyle 1\left|\begin{array}{cc}-2 & t-2\\-1 & 1\end{array}\right|+0+(t-3)\left|\begin{array}{cc}t-3 & -1\\-2 & t-2\end{array}\right|$=$\displaystyle -2+t-2+(t-3)[(t-3)(t-2)+2]=t-4+(t-3)[t^{2}-2t-3t+6+2]$=$\displaystyle t-4+(t-3)[t^{2}-5t+8]$
=$\displaystyle t-4+t^{3}-5t^{2}+8t-3t^{2}+15t-24=t^{3}-8t^{2}+24t-28$
there are some roots possible by theory t=1,-1,+28,-28
t=1: 1-8+24-28
t=-1: -1-8-24-28
from the calculator the other dont work too
??
• Oct 1st 2011, 05:25 AM
Deveno
Re: jordan form question
be very careful with your signs when working out determinants. for |tI - A|, i get:

$\displaystyle (-2)(1) - (t-2)(-1) + (t-3)[(t-3)(t-2) - (-1)(-2)]$

$\displaystyle = -2 + t - 2 + (t-3)[t^2 - 5t + 6 - 2]$

$\displaystyle = t - 4 + t^3 - 5t^2 + 4t - 3t^2 + 15t - 12 = t^3 - 8t^2 + 20t - 16$.

possible rational roots of this are 1,-1,2,-2,4,-4,8,-8,16,-16. i believe you will find this factors completely.