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?