(adsbygoogle = window.adsbygoogle || []).push({}); Q: Suppose the only eigenvalues of A are 1 and -1, and A is similar to a diagonal matrix. Prove that A^-1 = A

My Attempt:

Suppose the only eigenvalues of A are 1 and -1, and A is similar to a diagonal matrix.

=>A is invertible (since 0 is not an eigenvalue of A)

and there exists invertible P s.t.

(P^-1) A P = D is diagonal

=> A= P D (P^-1)

=> A^-1 = P (D^-1) (P^-1)

Now if I can prove that D = D^-1, then I am done. ButI am stuck right here.The trouble is that the eignevalues of A can have any number of multiplicities, so D can be diag{1,1,-1,-1,-1}, diag{1,-1,-1,-1,-1,-1}, etc., there are infinite number of possible D's, how can I prove that D = D^-1 is always true for these infinite number of different settings?

Can someone please help me?

Thanks a lot!

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# Homework Help: Eigenvalues & Similar Matrices

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