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Find the eigenvalues?

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In order to find eigenvalues, you need to solve the secular equation:
det|A-kI|=0; here A is your matrix, A={(a,-1);(1,a)}, I is the unit matrix, {(1,0);(0,1)}, k is the sought eigenvalue.
Matrix A-kI looks as follows:
A-kI={(a-k,-1);(1,a-k)} (a-k and -1 on the left, 1 and a-k on the right)
Its determinant is:
Let us equate the determinant to zero to find eigenvalues.
It is clear that two eigenvalues are complex, no matter what a.
k1=a-i; k2=a+i 
I hope this is as clear as it gets. If you have questions, please, ask.