Using the eigen value properties, it is known that complex eigen values are present in conjugates. So, if one eigen value is 5+3i, other must be 5-3i. Also from the eigen value properties, the product of the eigen values is equal to the determinant of the matrix. Thus for a 2x2 matrix, two eigen values: 5+3i, 5-3i.
So, (det A) = (5+3i)*(5-3i) = 25 + 9 = 34.