you are thinking too hard. Just try to test if the operator preserves linearity, meaning .
for a)Trace of a matrix is just adding the diagonal entries. is Trace(A*B)=Trace(A)*Trace(B). For example take Trace of first matrix is 2, trace of second matrix is 1 but A*B = whose trace is 1. so Trace(A*B)=1 which does not equal Trace(A)*Trace(B)=2, so Tr is not a linear transformation.
b) so the linear map given by , so AB under this map goes to but this isnt equal to A under the map, times B under this map which is so this map isnt linear.
u can do the rest.