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Find the eigenvalues?

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1 Answer

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:
(a-k)2+1
Let us equate the determinant to zero to find eigenvalues.
(a-k)2+1=0
It is clear that two eigenvalues are complex, no matter what a.
 
a-k=±√-1=±i;
k=a±i;
 
k1=a-i; k2=a+i 
 
I hope this is as clear as it gets. If you have questions, please, ask.