Let $A \in M_n(k)$ such that $A^3-A^2 + A - I_n =0$
1.) Prove that $A$ is invertible and determine $A^{-1}$
2.) Calculate $det(A)$
For the first question :
$A^3-A^2 + A - I_n =0$
$A(A^2-A + I_n) =I_n$
Hence $A^{-1}$ = $(A^2-A + I_n)$
For the 2nd question :
Assume that $x^3-x^2 + x - 1 $ is the minimal polynomial of $A$ .
which has 1 real root $x = 1$ which is the eigenvalue.
since the determinant is equal to the product of eigenvalues $det(A)=1$
However I'm not sure if my assumption is correct nor if i should prove that A is diagonalizable for the determinant to be equal to the product of the eigenvalues.
Thanks in advance.