So for the first transformation any matrix A where T(A)=Tr(A)=0 will be in the kernel. So what is necessary about the diagonal entries of A if Tr(A)=0?
For the range, remember the range is the subset of the codomain of T which has a pre-image in the domain. So for the first transformation the domain of T is U and the Codomain is . We must find the subset of that is "hit" by T. This means we must find all possible real numbers for which there exists a matrix A so Tr(A)=a. All such numbers make up the range.
Is it possible to find a matrix A so Tr(A)=a for any real number ? If I asked, find a matrix A so Tr(A)=5, could you find one?
You should be able to apply these ideas to the other transformation as well.
Hope this helps. :-)