Well, I don't know if there is an easier way than this, but I suspect

there is:

a) If you multiply Ax=b then you will get three equations for which

3 of the variables in x can be eliminated leaving you with

so

b) the kernel of A can be found in a similar way Ax=0

which is a straight line through the origin,

so a basis could be which has dimension 1, ie n(A)=1

c) I'm not sure what the rank theorem is but r(A)+n(A)=dimX means

that r(A)+1=4 so n(A)=3. But I thought that was kind of obvious anyway,

so maybe I misunderstood the question.