Is trans-1,2-dimethylcyclopropane optically active or not?
First off ...
There is a difference between &quot;chiral&quot; and &quot;optically active&quot;.  Something that's optically active has to have chiral molecules.  However, something can be chiral but if you have equal amounts of the mirror-image isomers, or if molecular motion can make an equal number of isomers like an umbrella-like inversion of a lone pair or an exchange of protons, then you do not see optical activity.  All we are looking at here is whether the molecule is chiral, not whether a sample of the stuff is in general optically active.
With that cleared up:
As Weijun says in the comments you have to have a mirror plane, or an improper rotation axis, or else the molecule is chiral.  A (proper) rotation axis does nothing in this regard.
Each of the carbon atoms with a methyl group on it is, in fact, a chiral center.  The four different groups attached to it are:
No 4.  A hydrogen atom
No 3.  The methyl group that's attached outside the ring
And...
No 2, No 1.  The ring going clockwise and going counterclockwise, as viewed from &quot;above&quot; the C–H bond, are counted as two different attached groups, as if the ring were &quot;cut off&quot; in either direction before rejoining the carbon atom you're looking at.
If you look at the ring attachment that way, you see that the two different directions rank 1 and 2, in some order depending on where the second carbon with the methyl group is, hence my numbering above.  So you can figure out an R or S configuration for each chiral center.  In the cis isomer you end up with one R and one S (thus mirror-image configurations), and a mirror plane does go between them making a meso rather than a chiral molecule.  But the trans isomers give both R or both S, which never allows a mirror plane or any improper rotation axis, and so they are chiral.