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. |