Suppose there is a matrix A such that A(adsbygoogle = window.adsbygoogle || []).push({}); ^{-1}= A. What can we say about the eigenvalue of A, g?

1) Ax= gx

2) A^{-1}Ax= A^{-1}gx

3) Ix= g A^{-1}x

4)x= g Ax

5)x= g gx

6) 1x= g^{2}x

Therefore

7) g^{2}= 1

8) g = 1 or g = -1

But suppose A = I (the identity matrix). For I, the only eigenvalue is g = 1 (g = -1 is not an eigenvalue of I). So, something must be wrong in the steps above. Can anyone point out what and where?

Thanks a lot...

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# Nonexistent eigenvalue

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