The key to obtaining a unique solution of the system is to find . What happens when does not exist?
Consider the system
In matrix form this becomes
Identify the matrix and hence find .
and Hence does not exist.
Looking at the original system we see that Equation (2) is simply Equation (1) with all coefficients doubled. In effect we have only one equation for the two unknowns and . The equations are consistent and have infinitely many solutions .
If we let assume a particular value, say, then must take the value i.e. the solution to the given equations is:
For each value of there are unique values for and which satisfy the original system of equations. For example, if , then , , if then , and so on.
Now consider the system
Since the left-hand sides are the same as those in the previous system then is the same and again does not exist. There is no solution to the Equations (3) and (4).
However, if we double Equation (3) we obtain
which conflicts with Equation (4). There are thus no solutions to (3) and (4) and the equations are said to be inconsistent .
What can you conclude about the solutions of the following systems?
First write the systems in matrix form and find :
Now compare the two equations in each system in turn:
The second equation is
times the first equation. There are infinitely many solutions of the
form where is arbitrary.
If we multiply the first equation by
which is in conflict with
the second equation. The equations are inconsistent and have no solution.