You're trying to confirm that this is not true in general? Not sure why the counter example you have provided is invalid. A=(2,2,2,;1,2,3) has row rank 2, but if you add a row of (1,1,1) its obvious that it doesn't have row rank of 3.
Does in general case, If i have (m x n) matrix A with full row rank rk(A)=m and add one more row (1,..,1) I get matrix A* with rk(A*)=m+1?
It does not seems true to me as one of the A rows can be k*(1,..,1), where k is real and then rk(A*) is not m+1. But I am asked to confirm that.