Example 1.  Find the nontrivial solutions to the homogeneous system  
        

Solution 1.

Use Gaussian elimination to eliminate    and the result is  

        

Since the third equation is a multiple of the second equation, this system reduces to two equations in three unknowns:

          

We can select one unknown and use it as a parameter.  For instance, let  ;  then the second equation implies that  and the first equation is used to compute .  Therefore, the solution can be expressed as the set of relations:

          
    or
        

We can fin the solution by entering 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.

Find the reduced row echelon form of the augmented matrix  M = [A, B].  

The equation form for this matrix is  

        

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

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 !

(c) John H. Mathews 2004