Example 5.  Balance the propane-oxygen equation by solve the homogeneous linear system

        

Solution 5.

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.

Find 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 three equations and four 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 4×4 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 !

(c) John H. Mathews 2004