Find B such that $\det(B)>0$ but there is no real $A$ with $\exp{(A)}=B$

Question

I have proved that $\exp(\text{tr}(A)) = \det (\exp (A))$. An easy corollary is that $\exp (A)$ has positive determinant whenever $A$ is real. Now the question asks us to find a B such that $\det(B)>0$ but there is no real $A$ with $\exp{(A)}=B$. I have no idea how to proceed so it would be greatly appreciated if someone could drop me a hint.