Example 2.  Solve the linear system of equations  
            

Solution 2.

Enter the equations into Mathematica.  

Identify the matrix of coefficients A and column vector B for the matrix problem AX = B.  

Form the augmented matrix  M = [A, B]  and perform Gauss-Jordan elimination with row interchanges.

This time Gauss-Jordan elimination encountered division by error and could not find a solution.  
We might have suspected a problem because the determinant of  A  is zero.

Let us investigate further and the reduced row echelon form of the augmented matrix  M = [A, B].  

This linear system is equivalent to:

The third equation states that   which is a contradiction.
In this case we say that the system of equations is inconsistent and there is no solution.  

We are done.

Aside.  We can let Mathematica find the reduced row echelon matrix.  This is just for fun !

Aside.  We check out  .  

Looking at the above calculations we see that    and  .    
Since    the theorem states that the system is inconsistent and has no solution.  

(c) John H. Mathews 2004