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I still remember the following question on my linear algebra test in college:

"Prove that A and A-Inverse have the same Eigen Values and corresponding Eigen Vectors."

The solution to this made no sense, but I managed to memorize it so that I could pass the test.

Forest through the trees?



It's not quite true, is it...?

  Av = cv        v is an eigenvector, c the corresponding eigenvalue
  (A^-1)Av = (A^-1)cv
  v = c(A^-1)v
  (1/c)v = (A^-1)v
So if c is an eigenvalue of A, then 1/c is an eigenvalue of (A^-1). (c can't be 0 because A is invertible, I think.)




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