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## Homework Statement

Prove that if the eigenvalues of a diagonalizable matrix are all + or -1, then the matrix is equal to its inverse.

i) Let D = P

^{-1}AP, where D is a diagonal matrix with + or -1 along its main diagonal.

ii) Find A in terms of P, P

^{-1}, and D.

iii) Use the fact that D is the diagonal and the properties of the inverse of a product of matrices to expand to find A

^{-1}.

iv) Conclude that A

^{-1}= A.

## Homework Equations

## The Attempt at a Solution

D * P

^{-1}= P

^{-1}AP *P

^{-1}

P * D * P

^{-1}= P * P

^{-1}A

PDP

^{-1}= A

Not sure if I'm heading in the right direction. I am drawing a blank here.