CHAPTER 05.12: SYSTEM OF EQUATIONS: If a solution exists, how do we know if it is unique?      In this segment we will talk about that is we have a system of equations and a solution exists-How do we know if it is unique? So let’s suppose somebody gives us a set of equations. AX=C And we have so far demonstrated that we can make a distinction between a system of equations being consistent or inconsistent. This is based on the fact that whether the rank of A is equal to rank of the augmented matrix AC. So we will have a consistent system of equations if the rank of the coefficient matrix is the same as the rank of the augmented matrix. Augmented meaning that we are putting the right side vector as the last column with the A matrix. If the rank of the two matrices is the same then we have a consistent system set of equations. But if the rank of A is less than the rank of the augmented matrix then we know that we have an inconsistent system of equations.   So what that means is that those are the only two possibilities which you are going to have and then-but in the inconsistent system set of equations we have two possibilities. One possibility is you are going to have a unique solution because consistent system of equations means that you have a solution. It doesn’t tell you whether you have one solution or infinite solutions. So there are two possibilities: unique solution or you are going to have infinite solutions. So the question rises-How do we make a distinction once we have established that particular system of equations is consistent-that the rank of the coefficient matrix is same as the rank of the augmented matrix-how do we make a distinction whether we are going to have a unique solution or infinite solutions? For that all we have to do is check if rank of A is equal to the number of unknowns. So if you find out that the rank of the coefficient matrix is the same as the number of unknowns then it has a unique solution and if we find that the rank of A-that is the coefficient matrix- is less than the number of unknowns then we have infinite number of solutions.   So those are the possibilities which we have to look at once we establish a particular system of equations is consistent or inconsistent by checking whether the rank of  the A matrix is the same as the rank of the augmented matrix or if the rank of the A matrix is less than the rank of the augmented matrix. Then once we have the rank of A coefficient matrix is the same as the rank of the augmented matrix to distinguish whether it is a unique solution or infinite solutions is based on if the rank of the coefficient matrix matches the number of unknowns or if the rank of the coefficient matrix is less than the number of unknowns. And that is the end of this segment.