CHAPTER 05.14: SYSTEM OF EQUATIONS: Does a set of equations have a unique solution? Example 2     In this segment we want to see if a set of equations has a unique solution or not we’ll take an example so somebody’s given you three equations three unknowns this is the quotient matrix, this is the solution vector, this is the right hand side vector so if you write in the matrix form the shortened symbolic matrix form we have [A][X] equal to [C] so what we are going to check is - what we are being asked is that - hey can you tell us whether it has a unique solution so the first thing which I will do is I want to find rank of (A) so I’m going to find the rank of the quotient matrix and it turns out to be 2 you have to work it out to see why the rank of (A) is 2 so it turns out to be 2 the rank of the augmented matrix also turns out to be 2 the augmented matrix is simply taking the right hand side vector adding it as the fourth column to the [A] matrix and then finding the rank of that three rows and four column matrix and it turns out to be 2 so what you are finding out is that rank of (A) is same as the rank of the augmented matrix which is 2 this simply means that it is a consistent system of equations so this set of equations which we have here is consistent what do we mean by consistent simply that it has a solution either the solutions are the number of solutions is 1, is unique, or it is infinite so we have to now make a decision based on this what we have found now that this is a consistent system of equations hey do we have unique solutions, unique solution, or do we have infinite solutions so how are we going to do that we are basically going to check what the rank of (A) is and we are going to check the number of unknowns we are finding out that the rank of (A) is 2 and the number of unknowns is 3 because 1,2,3 so the rank of (A) in this case is less than the number of unknowns so since that is the case we have infinite solutions. And that’s the end of this segment.