CHAPTER 05.14: SYSTEM OF
EQUATIONS: Does a set of equations have a unique solution? Example 2 In
this segment we want to see if a set of equations has a unique solution or
not we’ll take an example so somebody’s given you three equations three
unknowns this is the quotient matrix, this is the solution vector, this is
the right hand side vector so if you write in the matrix form the shortened
symbolic matrix form we have [A][X] equal to [C] so what we are going to
check is - what we are being asked is that - hey can you tell us whether it
has a unique solution so the first thing which I will do is I want to find
rank of (A) so I’m going to find the rank of the quotient matrix and it turns
out to be 2 you have to work it out to see why the rank of (A) is 2 so it
turns out to be 2 the rank of the augmented matrix also turns out to be 2 the
augmented matrix is simply taking the right hand side vector adding it as the
fourth column to the [A] matrix and then finding the rank of that three rows
and four column matrix and it turns out to be 2 so what you are finding out
is that rank of (A) is same as the rank of the augmented matrix which is 2
this simply means that it is a consistent system of equations so this set of
equations which we have here is consistent what do we mean by consistent
simply that it has a solution either the solutions are the number of
solutions is 1, is unique, or it is infinite so we have to now make a decision
based on this what we have found now that this is a consistent system of
equations hey do we have unique solutions, unique solution, or do we have
infinite solutions so how are we going to do that we are basically going to
check what the rank of (A) is and we are going to check the number of
unknowns we are finding out that the rank of (A) is 2 and the number of
unknowns is 3 because 1,2,3 so the rank of (A) in this case is less than the
number of unknowns so since that is the case we have infinite solutions. And
that’s the end of this segment. |