Can a Linear Operator Satisfying A^2 - A + I = 0 Always Have an Inverse?

[seek]

Homework Statement

Show that if an operator A satisfies A² - A + I = 0 then A has an inverse. Express A^-1 as a simple polynomial of A.

Homework Equations

I'm not sure that this is relevant, but A^-1=1/(detA)TrC where TrC is the transpose of the matrix of cofactors. Also:
If detA = 0 then the matrix has no inverse

The Attempt at a Solution

So I notice immediately that adding by the identity matrix in this equation will result in a matrix with its diagonal having numbers (real or complex) and the rest being zero, as I can be expressed as the kronecker delta. And if the determinant must be nonzero in order to have an inverse, there has to be a way to relate the diagonal of an n dimensional matrix with its determinant. I'm just stuck as to how to do that. Any help greatly appreciated, thank you.
*Edit:
I've been thinking more about this problem, and it seems like there should be a way to use the secular equation to solve it. We went over it briefly in class (the class is quantum and I haven't had linear algebra yet, so it's kind of a chore), but not in enough detail that I would be able to use it in a proof.

 

[tiny-tim]

Welcome to PF!

Hi seek! Welcome to PF!

So I notice immediately that adding by the identity matrix in this equation will result in a matrix with its diagonal having numbers (real or complex) and the rest being zero, as I can be expressed as the kronecker delta. And if the determinant must be nonzero in order to have an inverse, there has to be a way to relate the diagonal of an n dimensional matrix with its determinant. I'm just stuck as to how to do that. Any help greatly appreciated, thank you.
*Edit:
I've been thinking more about this problem, and it seems like there should be a way to use the secular equation to solve it. We went over it briefly in class (the class is quantum and I haven't had linear algebra yet, so it's kind of a chore), but not in enough detail that I would be able to use it in a proof.

oooh … so complicated …

Try writing it A² - A = -I ​

 

[seek]

My oversight is to my pride as a cold slap to the visage. Thanks so much for the help, maybe next time I'll be able to use the skills I learned in 5th grade.