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In an earlier paper published in the Journal of Natural Sciences Research in 2015 on how to balance chemical equations using matrix algebra, Gabriel and Onwuka showed how to reduce the resulting matrix to echelon form using elementary row operations. However, they did not show how elementary row operations can be used in reducing the resulting echelon matrix to row reduced echelon form. We show that the solution obtained is actually the nullspace of the matrix. Hence, the solution can be infinitely many. In addition, we show that instead of manually using row operations to reduce the matrix to row reduced echelon form, software environments like octave or Matlab can be used to reduce the matrix directly. In all the examples presented in this paper, we reduced all matrices to row reduced echelon form showing all row operations, which was not clearly stated in the Gabriel and Onwuka paper. Most importantly, with the availability of Mathematical software, we show that we do not need to carry out these row operations by brute force.

According to Risteski [

If the number of atoms of each type of element on the left is the same as the number of atoms of the corresponding type on the right, then the chemical equation is said to be balanced [

The algebraic approach discussed in [

In the next section, we state two well known results partaining echelon form and row reduced echelon form.

In this section, we state well known results about echelon form and row reduced echelon form. We will not bother about the algorithm as this is readily available in most Linear Algebra textbooks.

Lemma 2.1.: The number of nonzero rows and columns are the same in any echelon form produced from a given matrix

Given an

1. Use Gauss elimination to produce an echelon form from

2. Use the bottom-most non zero entry

Definition 2.1 An

1. It is in echelon form (with

2. The

The next result which can be found in [

Theorem 2.1 (Row Reduced Echelon Form): Each matrix has precisely one row reduced echelon form to which it can be reduced by elementary row operations, regardless of the actual sequence of operations used to produce it.

Proof. See [

Example 3.1.: Rust is formed when there is a chemical reaction between iron and oxygen. The compound that is formed is a reddish-brown scales that cover the iron object. Rust is an iron oxide whose chemical formula is

Balance the equation.

In balancing the equation, let

We compare the number of Iron (Fe) and Oxygen (O) atoms of the reactants with the number of atoms of the product. We obtain the following set of equations:

The homogeneous system of equations becomes

From the above, the matrix

must be one. Hence, we replace row two with half row two, that is

Thus,

Upon expanding, we have

the nullspace solution

There are three pivot variables

or

Example 3.2.: Ethane

balance the equation.

Let the unknowns be

We compare the number of Carbon (C), Hydrogen (H) and Oxygen (O) atoms of the reactants with the number of atoms of the products. We obtain the following set of equations:

In homogeneous form,

In the first step of elimination, replace row two by row two minus three times row one, i.e.,

Exchange row two with row three or vice versa to reduce

In the next set of operations that we will carry out to reduce

The last operation that will give us

The solution to

Upon expanding, we have

the nullspace solution

There are three pivot variables

free variable

Example 3.3.: Sodium hydroxide (NaOH) reacts with sulphuric acid

Balance the equation.

In balancing the equation, let

We compare the number of Sodium (Na), Oxygen (O), Hydrogen (H) and Sulphur (S) atoms of the reactants with the number of atoms of the products. We obtain the following set of equations:

Re-writing these equations in standard form, we have a homogeneous system

or

The augmented system becomes

Since the right hand side is the zero vector, we work with the matrix

Replace row 2 with row two minus row one i.e.,

In the second set of row operations, we replace row three by two times row three minus row two or

In the third stage of the elimination process, we replace row four with 3 times row four plus row three i.e.,

We now reduce

Replace row one by row one plus two times row three i.e.,

replaces all nonzeros above the pivots to zero resulting in the row reduced echelon form

The solution to

Upon expanding, we have

the nullspace solution

There are three pivot variables

Example 3.4.: Using row reduced echelon form, balance the following chemical reaction:

Let

We obtain the following set of equations for each of the elements:

The corresponding matrix becomes

The following row operations

In the same vein, the following row operations

Finally,

There are three pivots respectively

The row operations

Therefore, the solution

For simplicity, we equate

In this section, we use octave to reduce each of the matrices considered in the last section to row reduced echelon form. We remark that just as predicted by the theory, row exchanges does not change the outcome of row reduced echelon form. This means that if you interchange any of the row of each of the matrices in the four examples, the rref will be the same.

Example 4.1.: Type the matrix

Example 4.2.:

Example 4.3.:

Example 4.4.:

In the next example, we illustrate the power of the rref command.

Example 4.5.: Consider balancing the following chemical reaction from [

Let the unknown coefficients be

We write down the balance conditions on each element as

Sodium:

Chlorine:

Sulphur:

Oxygen:

Hydrogen:

After transposing, the above system of equations can be written in the form

Using Matlab or Octave

If we set

In this paper, we have shown how to balance chemical equations using row reduced echelon form. In actual fact, the echelon form alone could have been used and we still have the same solution but reducing it to rref makes the solution easily deduced. This paper improves on the work of Gabriel and Onwuka and we show that the octave/ Matlab rref command can be used to confirm the correctness of the final output on the one hand or as a stand alone.

Akinola, R.O., Kut- chin, S.Y., Nyam, I.A. and Adeyanju, O. (2016) Using Row Reduced Echelon Form in Balancing Chemical Equations. Advances in Linear Algebra & Matrix Theory, 6, 146- 157. http://dx.doi.org/10.4236/alamt.2016.64014