Balancing Chemical Equations Using Matrices

Early on in your chemistry studies, you will have ample opportunity to balance equations! This is a fundamental skill in chemistry, as you might have noticed from the short reading in stoichiometry! Balancing equations means writing chemical equations such that the amount of stuff you start with in the reaction equals the amount of stuff you end up with as a product.
For example:

[A]Fe₂O₃ + [B]Al → [C]Al₂O₃ + [D]Fe

The blue letters represent the unknown coefficients to the balanced equation. To solve for these unknowns a system of equations must be generated. The easiest way to do this is to write a matrix relating the quantity of each element found in each reagent. The following table represents a break-down of this process, where each row represents a different element and each column represents an unknown coefficient. In this reaction there are 3 elements involved and 4 unknown coefficients.

	Fe2O3	+	Al	=	Al2O3	+	Fe
Fe	2		0		0		1
O	3		0		3		0
Al	0		1		2		0

Alternatively, this can be represented as a system of linear equations:

2A + 0B = 0C + 1D
3A + 0B = 3C + 0D
0A + 1B = 2C + 0D

Solve this system and you will easily get the answer !

Data retrieved from: http://www.personal.psu.edu/jzl157/balance.htm