You ve complicated it a bit.

It has to be summed over j and k. Now in cartesian coordinates, these indices take values of 1, 2 and 3. Replace the index of j and k accordingly and you get a1x1+a1y1 + a2x2+a2y2 + a3x3+a3y3.

You ve confused things by introducing rows and columns which is wrong. Just replace the indices with these values and if there is dummy index, sum over that. Here it is a simple expansion.