Example 3.  Solve the homogeneous linear system of equations  
          

Solution 3.

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 find its reduced row echelon form.

The equation form for this matrix is  

      

There are two free variables which we choose to be    and  .   They is used in computing  .  

Solve the previous equations for  .  

      

Make the substitutions    and  .  

    
    
Get    
    

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 two rows are entirely zero, the system has reduced to one equations and three unknowns.  

We can add the equations    and  . 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 can graph the situation.  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