If $X$ is a random variable distributed as a Gumbel with location $\gamma$ and scale $1$ (where $\gamma$ is the Euler constant)
[Hence the mean is $$ \gamma-1\times\gamma=0 $$ and the variance is $$ \pi^2/6\times 1^2=\pi^2/6 $$]
(1) What is the distribution of $aX$ where $a\in \mathbb{R}$? $a$ should be $>0$?
(2) What is the distribution of $-X$?
I don't know if it is relevant, but here I'm using the definition of Gumbel as in Mathworld and Matlab (sign flipped with respect to the definition of Gumbel in Wikipedia)
Still a Gumbel distribution.
As with Normal distributions, the family of Gumbel distributions includes all pdfs of the form $|\beta|^{-1}f\left(\frac{x-\mu}{\beta}\right)$, where $f$ is in the family. So $X\mapsto aX$ just scales $\beta$ to $a\beta$, including in the case $a=-1$.