(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that if an operator A satisfies A^{2}- A + I = 0 then A has an inverse. Express A^{-1}as a simple polynomial of A.

2. Relevant equations

I'm not sure that this is relevant, but A^{-1}=1/(detA)TrCwhere TrCis the transpose of the matrix of cofactors. Also:

If detA = 0 then the matrix has no inverse

3. 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, asIcan 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.

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# Homework Help: Inverse of a linear operator

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