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. |