CHAPTER 05.13: SYSTEM OF
EQUATIONS: Does a set of equations have a unique solution? Example 1 In
this segment we will see if a set of equations has a unique solution. Let’s
take an example. We are being asked that: Hey, you are given a set of
equations, can you check whether this has a unique solution? 25, 5, 1, 64, 8,
1, 144, 12, and 1. Unknowns are X1 X2 and X3. This is 106.8, 177.2, and 279.2.
So what we have to do is to check whether it has a unique solution or not.
The first thing which we have to do is check-if we call this A –the
coefficient matrix. X is the unknown vector. C is the right hand side. The
first thing that we have to do is check the rank of A. The rank of A is to be
found as 3. So of course I am just telling you the rank of A is 3, you have
to work it out, that the rank of A is 3. The rank of the augmented matrix,
which is putting this as the 4th column in the matrix, is also found to be 3.
And then the number of unknowns, which we can see here: 1 2 3, is 3. Because
the rank of the coefficient matrix is the same as the rank of the augmented
matrix, this means that, this is a consistent set of equations. Which means
that either it has a unique solution or it has infinite number of solutions.
But since the rank of A is the same as the number of unknowns that implies
that it has a unique solution. And that is how we are able to figure out if a
system of equations has a unique solution or not. And that’s the end of this
segment. |