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.

 As the mineral known as hematite, Fe2O3 is the main source of the iron for the steel industry. Fe2O3 is paramagnetic, reddish brown, and readily attacked by acids. 

Aluminium is the third most abundant element (after oxygen and silicon), and the most abundant metal, in the Earth's crust. It makes up about 8% by weight of the Earth's solid surface. Aluminium metal is so chemically reactive that native specimens are rare and limited to extreme reducing environments. Instead, it is found combined in over 270 different minerals.[5] The chief ore of aluminium is bauxite.

Aluminium oxide is a chemical compound of aluminium and oxygen with the chemical formula Al2O3. It is the most commonly occurring of several aluminium oxides, and specifically identified as aluminium(III) oxide.

Iron is a chemical element with the symbol Fe (from Latin: ferrum) and atomic number 26. It is a metal in the first transition series. It by mass is the most common element on Earth, forming much of Earth's outer and inner core. It is the fourth most common element in the Earth's crust.


Alternatively, this can be represented as a system of linear equations:
2A + 0B = 0C + 1D
3A + 0B = 3C + 0D
0A + 1B = 2C + 0D