Example 2.  Solve the homogeneous 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:

There is one free variable which we choose to be  .  It is used in computing  .  

Solve the previous equations for   .  

      

Make the substitution  .

      

The solution vector    is

We are done.

Aside.  We can verify that this is the solution by direct multiplication A X.  This is just for fun !

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

Notice.  Since the last row is entirely zero, the system has reduced to two equations and three unknowns.  

We can add the equation    to those in the reduced row echelon form and then row reduce one more time to get the solution.

The 3×3 identity matrix appears in the left 3 columns of  M, and the given linear system is equivalent to:

The solution vector is the fourth column of  M.

We can verify that this is the solution by direct multiplication A X.  This is just for fun !

We are really done.

Aside.  We might wonder if Gauss-Jordan elimination could be use to find some solution, after all there is an infinite number of them.

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

So we have found one of the infinite number of solutions to be  .

Notice that this is merely substituting t=1 in the general solution  .  

Aside.  We can graph general solution.  This is just for fun.  

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 the theorem guarantees that the system has an infinite number of solution.    However, it might be easier to just check out the determinant.  

(c) John H. Mathews 2004