Example 4.  Solve the linear system  
          

Solution 4.

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  .    
Since    the theorem guarantees that there will be an infinite number of solutions.  

We are really really done.

Aside.  The following Mathematica steps will also solve the problem automatically.  It starts with the equations, creates the matrices, and ends up with the vector form of the solution.  This is just for fun !

(c) John H. Mathews 2004