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Let $A$ be a real $2\times2$ matrix. If $5+3i$ is an eigenvalue of $A$, the $\det(A)$

a. equals $4$

b. equals $8$

c. equals $16$

d. cannot be determined from the given information

$\mathbf{My\ Approach}$

Since $A$ is a real matrix, it will give a quadratic characteristic equation with real coefficients with $5+3i$ being one of the roots. Therefore, the second eigenvalue has to be $5-3i.$ Hence $\det(A)=(5+3i) \cdot (5-3i)=25+9=34.$

Which is not an option. Where am I going wrong?

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    $\begingroup$ You have a typo in this. The second eigenvalue is $5-3i$. But you are correct, and none of their answers is correct. $\endgroup$
    – Ted Shifrin
    Feb 28, 2017 at 17:00
  • $\begingroup$ My apologies. Thank you! $\endgroup$
    – Naweed G. Seldon
    Feb 28, 2017 at 17:02
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    $\begingroup$ Yes all your theory is right. $\endgroup$
    – mathreadler
    Feb 28, 2017 at 17:03
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    $\begingroup$ Most likely the author made a sign error: $25 - 9 = 16$. So option c is probably the "correct" option. $\endgroup$
    – Reinstate Monica
    Feb 28, 2017 at 17:09
  • $\begingroup$ @TedShifrin It does not say anywhere that at least one should be correct. $\endgroup$
    – mathreadler
    Mar 16, 2017 at 16:39

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