Eigen values always occur in conjugate pairs. So det(A)=(5+3i)*(5-3i)=34
EDIT: It is wrong to say that “Eigen values always occur in conjugate pairs”. What makes them appear in conjugate pair is the fact that given matrix is a real matrix. Why?
It is because determinant of real matrix is a real number and the determinant is equal to product of all the eigen values.
$\lambda _{1}$$\lambda _{2}$$\lambda _{3}$$\lambda _{4}$….. = determinant
If determinant is real number only then the product $\lambda _{1}$$\lambda _{2}$$\lambda _{3}$$\lambda _{4}$… is forced to be a real number. This implies the number of imaginary eigen values should be even i.e. they must occur in pairs. Occuring even number of times is still not sufficient to say that imaginary part will become zero- take for example (2+i)*(3+5i)-->imaginary part is $13i$. This means there is something more to it so that product has imaginary part as zero. That thing is “conjugate pairs”- take for example (2+i)*(2-i)-->imaginary part is zero. If determinant is real number then eigen values must occur in conjugate pairs can also be observed from this equation :- $\lambda$$=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}$
Therefore, since the matrix is real, the eigen values occur in pairs...and moreover in conjugate pairs. Hence second eigen value must be 5-3i
