Determinant of a $2\times 2$ real matrix when an eigenvalue is given

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?

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