If we have more equations than unknowns, does it mean we have inconsistent system of equations?

CHAPTER 05.15: SYSTEM OF EQUATIONS: If we have more equations than unknowns, does it mean we have inconsistent system of equations?

In this segment we’ll answer the question that many times you might have more equations than unknowns or more unknowns than equations. So in that case, does it mean that we have inconsistent system of equations? Because most times people think that whenever you are given simultaneous linear equations you have n number of equations and n number of unknowns. But that’s not necessarily the case; you could have more unknowns than equations or more equations than unknowns. So let’s answer this particular question that if we have more equations than unknowns does it mean that we have inconsistent system of equations. The categorical answer is no. Let’s think of some examples and see what happens.

So we’ll take key examples, so let’s take this example right here. So if somebody gives us a set of equations like this: 25, 5, 1, 64, 8, 1, 144, 12, 1. 89, 13, 2 and X1, X2, X3 is equal to 106.8, 177.2, 279.2 and 284.0. So here we are clearly seeing that we got four equations and three unknowns, because you got four rows right here and four rows right here. And a few might say hey I cannot solve this set of equations so we don’t know. So what we have to see is that whether the question or whether the particular system of equations is consistent or inconsistent. It has to be looked at because it’s all based on the rank of the caution matrix and the rank of the augmented matrix. In this case so if we write this symbolically that hey this is AX equal to C, what we want to see is that the rank of A, and this again you have to find out, the rank of A is three. And you will also find out that the rank of the augmented matrix which will be just adding this column to the A matrix here, you’re going to find out hey that is also three.

And since you’re finding out the rank of the caution matrix is the same as the rank of the augmented matrix what does that mean? That means that we have a consistent system of equations. Not only that since the rank of A, is equal to the number of unknowns. Because the rank of A is three and the number of unknowns is also three. X1 X2 and X3. So what that also means is that it has a unique solution. So not only are we able to show that somebody gives us four equations three unknowns that we have a consistent system of equations for this particular case that we can have either infinite solutions or unique solution. But then by comparing the rank of the caution matrix with the number of unknowns which is the same number, we’re also able to show that this particular system of equations, four equations and three unknowns has a unique solution.

So let’s look at another example. Let’s look at another example. Somebody gives me a set of equations like this one: 25, 5, 1, 64, 8, 1, 144, 12, 1, 89, 13, 2. X1, X2, X3 is equal to 106.8, 177.2, 279.2, and 280.0. So somebody is now giving you again four equations and three unknowns, so we wanted to remind whether this particular system of equations is consistent or inconsistent. SO in this case what you will find out is when you run this process you will find out the rank of A will turn out to be 3. The rank of A will turn out to be three. So in this caution matrix, so let me write this as A times X is equal to C. So you got A as the caution matrix, X as the unknowns vector and C as the right hand side vector. You’ll find the rank of this caution matrix it turns out to be three, and the rank of the augmented matrix which is basically this column being added to the A matrix, so making it four rows and four columns, that turns out to be four. So what we are finding out is that rank of A is less than the rank of the augmented matrix because this is three and this is four. So what does this mean? That means that we have an inconsistent system of equations, because the rank of the caution matrix is strictly less than the rank of the augmented matrix. So which means that when we say it’s inconsistent it means that there is no solution exists for this set of equations.

And we look at yet another example, to see that hey what does it mean when we have more equations than unknowns. Let’s look at another example, C, it’s the third example we’re taking. And somebody gives us this set of equations: 25, 5, 1, 64, 8, 1, 50, 10, 2, 89, 13, 2. X1, X2, X3 is equal to 106.8, 177.2, 213.6, and 280.0. So somebody is giving us four equations and three unknowns, and it’s asking us hey can you figure out whether this particular system of equations is consistent or inconsistent. So when you would look at this particular matrix, you’ll find out that the rank of, so let me just write it symbolically, AX equal to C. So A is my caution matrix, X is my unknown vector and C is my solution vector. We find out that hey rank of A turns out to be two. And of course you have to find it out why that is so, you’ll find out the rank of this caution matrix which is four rows and three columns turns out to be two. The rank of the augmented matrix which is taking this right hand side right here putting that as the fourth column here so we got a four by four matrix, and that rank also turns out to be two. So the rank of the caution matrix is the same as the rank of the augmented matrix. That means that we have a consistent system of equations. So that means either we have a unique solution or we have an infinite number of solutions.

But what we are finding out is that rank of A and if we compare them with the number of unknowns what do we see? The rank of A is two and the number of unknowns is three so rank of A is less than the number of unknowns which means that we although it is a consistent system of equations we’ll have infinite solutions. So although we’re asked to only figure out whether a particular system of equations is consistent or not but we’re going a step further to say hey if the system of equations turns out to be consistent hey do we have a unique solution or do we have infinite solutions. In this example the rank of A, same as the rank of the augmented matrix proving that hey it is a consistent system of equations, but the rank of the caution matrix is less than the number of unknowns and that tells us hey we’ll have infinite number of solutions. And that’s the end of this segment.