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Given a matrix $A = \begin{pmatrix} 40 & -29 & -11\\ \ -18 & 30\ & -12 \\\ \ 26 &24 & -50 \end{pmatrix}$ has a certain complex number $l\neq0$ as an eigenvalue. Which of the following must also be an eigenvalue of $A$: $l+20, l-20, 20-l, -20-l?$

It seems that complex eigenvalues occur in conjugate pairs. It is clear that the determinant of the matrix is zero, then $0$ seems to be one of the eigenvalues.

Please suggest.

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    @Tsotsi I suppose it is clear by hand calculation.2013-05-11

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Hint: The trace of the matrix is $40+30+(-50)$. As you observed, $0$ is an eigenvalue.

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    You are welcome. It is clear when you see it, but if one doesn't notice, it looks as if one needs to do the full determinant calculation.2013-05-11