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. |