1. The problem statement, all variables and given/known data The system Ax = b does not have a solution. A is a full column rank matrix. Multiply both sides of the equation, Ax =b with AT. We get, AT A x = AT b Solving for x now, we get x = [inverse of ( AT A)] ATb By using relevant examples, we find that solution for the system exists, a contradiction to what the system looked like originally! How is this possible? Is there some incorrect assumption? 3. The attempt at a solution One doubt that I have is that I am not entirely sure whether the operation of multiplying the system with AT on both sides from the left, is a valid one in the first place. inverse of AT does not exist. So there is no way of returning back to the original system i.e Ax = b from AT A x = AT b Is this a reasonable question and ,if not, where I am going wrong? What seems to be the problem?