Prove $2 \times 2$ real matrix is not the exponential of some other real matrix

Question

How do I prove that the real matrix $A$, where $$ A=\begin{pmatrix} -3 & 0\\ 0&-5 \end{pmatrix}$$ can not be written es the exponential of another real matrix?

Answer 1

A real $2\times2$ matrix $X$ has either two real eigenvalues or a conjugate pair of non-real eigenvalues.

If $X$ has two real eigenvalues, then $e^X$ has two positive eigenvalues and hence it isn't equal to $A$.

If $X$ has a conjugate pair of non-real eigenvalues, then $e^X$ has a conjugate pair of eigenvalues too and so it still cannot possibly be equal to $A$.