
determinant problem
$\displaystyle A=\left(\begin{array}{ccc}3 & 1 & 1\\7 & 5 & 1\\6 & 6 & 2\end{array}\right)$find the definative polinomial?
if i go by definition
$\displaystyle tIA=(t+3)[t^{2}3t4]+13t+20$
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 $\displaystyle (t+2)^{2}(t4)$
is there some other way to get this result
?

Re: determinant problem
check your arithmetic.
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).

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

Re: determinant problem