Proving a statement about determinants

Since $A$ has no real eigenvalues yet is a real matrix, all its eigenvalues must come in complex conjugate pairs $a\pm bi$ with $b\ne0$. But $(a+bi)(a-bi)=a^2+b^2>0$, so the product of all eigenvalues of $A$ – its determinant – must be positive. Hence $\det A=1$.