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Let the matrix $$ A = \begin{bmatrix} 40 & -29 & -11 \\ -18 & 30 & -12 \\ 26 & 24 & -50 \\ \end{bmatrix}$$ have a certain complex number $p \neq 0$ as an eigenvalue. Which of the following must also be an eigenvalue of $A$?

  1. $p+20$
  2. $p-20$
  3. $20-p$
  4. $-20-p$
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Can you tell us why you cannot solve this? This should be quite standard. –  Bombyx mori Nov 30 '12 at 5:58
@user32240 Also being an Indian, I can tell that this is a question from some competition and she wants be sure if her solution is correct. –  Dilawar Nov 30 '12 at 6:01
Are those column vectors , row vectors? –  diimension Nov 30 '12 at 6:04
@diimension I don't think it really matters in this context, the eigenvalues are invariant under transpose. –  EuYu Nov 30 '12 at 6:13
@diimension As do we all :) –  EuYu Nov 30 '12 at 6:35
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2 Answers

Hint: $0$ is an eigenvalue, and the sum of the eigenvalues is the trace.

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Add column 2 and column 3. You will notice that sum is negative of column 1. Thus the matrix's determinant is 0. When the determinant of a matrix is zero, what we can say about one of it's eigenvalue?

Then go for trace of a matrix and its relation with eigenvalues of a matrix.

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