1. ## determinant problem

$A=\left(\begin{array}{ccc}-3 & 1 & -1\\-7 & 5 & -1\\-6 & 6 & -2\end{array}\right)$find the definative polinomial?
if i go by definition
$|tI-A|=(t+3)[t^{2}-3t-4]+13t+20$
so i need to open the cols and find the roots of polinomial of power 3 which is alot of work
the answer in the book is $(t+2)^{2}(t-4)$
is there some other way to get this result
?

2. ## Re: determinant problem

for det(tI - A), i get (t+3)(t^2 - 3t - 4) + t - 4 = t^3 - 12t - 16.

to factor t^3 - 12t - 16, you might try guessing rational (and thus integer) roots first, that is: t = -1,1,-2,2,-4,4,-8,8,-16,16.

on the 3rd try, you would find that (-2)^3 - 12(-2) - 16 = -8 + 24 -16 = 0, so you would know that t+2 is a factor.

division by t+2 will give you a quadratic, to which you could apply the quadratic formula (unless you could factor it by inspection).

3. ## Re: determinant problem

Noticing that "t+ 2" factor, I would have been inclined to expand the determinant on the third column!

ok thanks