CHAPTER 05.08: SYSTEM OF EQUATIONS: Distinguishing between consistent and inconsistent system of equations based on rank of matrices

In this segment we will want to see that, hey, if somebody gives us a system of equations, are we able to figure out whether that particular equation is consistent or inconsistent? I’m going to base it on now the definition of the rank of matrices. So we are no longer trying to do this by visual inspection, which won’t be possible if we have many, many equations and many, many unknowns. So we want to be able to figure out a particular set of equations is consistent or inconsistent. Consistent meaning that there is a solution whether that it’s unique or infinite. That’s what we mean by consistent equation. Inconsistent means that there is no solution at all.

Let’s go and see how that is related. So somebody gives us A X = to C. Somebody gives us a system of equations, saying A times X is equal to C. Then what they want us to figure out- is this a system of equations that is consistent or inconsistent? So if I need to make a decision whether it is consistent that will be based on the rank of A is equal to the rank of the augmented matrix. So what we mean by augmented matrix is that we take the right side vector and add that as a column to the A matrix. So the A matrix let’s suppose has 3 rows and 3 columns and this one of course will have one column. The augmented matrix will turn out to be 4 columns and 3 rows. So the augmented matrix-now we have the rank of the augmented matrix - we calculate the rank of the coefficient matrix. If they are equal then we have a consistent system of equation which means we either have a unique solution or we have infinite solutions.

How do we determine whether we have-if we are able to find out that a system of equations is consistent? Now how do we figure out if it has a unique solution or infinite solutions? That’s something we will do in a separate segment. Now this one will be inconsistent. So we will have no solution at all if we find that the rank of A is less than the rank of the augmented matrix. So the procedure is very simple, if you want to figure out whether a particular system of equations is consistent or inconsistent. You calculate the rank of the coefficient matrix and you calculate the rank of the augmented matrix, if the 2 ranks are the same then it is consistent. If the rank of the coefficient matrix is less than the rank of the augmented matrix then it is inconsistent. And we will illustrate this phenomenon through examples. And that’s the end of this segment.