Showing that a matrix cannot be obtained as an element of a one parameter subgroup of $SL(2; R)$

Question

I have to show that the matrix \begin{bmatrix} -\lambda & 0 \\ 0 & -\frac{1}{\lambda} \end{bmatrix}

cannot be obtained as an element of a one parameter subgroup of $SL(2; R)$ except when $\lambda = 1$. Additionally, show that it can be obtained as a combination of paths $e^Ae^B$ and find $A$ and $B$.

I don't even know how do I start proving this result. Given a one parameter subgroup, I can verify versus some element or list all the elements but I need help with this.

Answer 1

Hint: eigenvalues. What are the eigenvalues of $\exp(tA)$ in terms of the eigenvalues of $A$?