I'd like to write to you two problems that I tried to solve, I'm not sure of the solution of the first.
Let $A\in M_n (F)$ be a matrix and $g(t)\in F(t)$ a polynomial, $P_A(t)$- the characteristic poly of $A$. I need to prove that if $\gcd(P_A(t),g(t)=1$ so $g(A)$ is invertible.
Things I know:
*If $\gcd(P_{A}(t),g(t)=1$ so $P_A(t)$ and $g(t)$ have no factor in common
*for every eigenvalue $\lambda$ of A $\det(A-\lambda)=0$
*The minimal poly has all the factors that the cha. has and it is the poly with the minimal degree that if we plug $A$ into it it will equal 0 So I know that for sure $g(A) \neq0$ , but it's not enough to say that $\det g(A) \neq 0$ and to conclude what I need.
Is it correct to say that if $\det g(A) = 0$ than $g(x)$ was part of $P_A(t)$? And if yes, why?
The second question is for this bilinear form: $q(v)=q(x,y,z)=2x^2-3y^2+xy-5yz$ I need to find base $B$ so that $[q]^B$ will be diagonalizable matrix. So, I tried to look for eigenvalues but it's impossible mission, it's really messy. Is there any other method which with I can find it? maybe something with Jacoby method?
Thanks alot!