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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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    @diimension As do we all :)2012-11-30

2 Answers 2

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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.