While working with systems of linear equations in the 1850s and 1860s, the English mathematician Arthur Cayley developed the use of a matrix (as we will show, a matrix is simply a rectangular array of numbers).

His interest was in the mathematical methods involved, and for many years matrices were used by mathematicians with little or no reference to possible applications other than solving systems of equations.

It was not until the 1920s that one of the first major applications of matrices was made when physicists used them in developing theories about the elementary particles within the atom. Their work is still very important in atomic and nuclear physics.

Since about 1950, matrices have become a very important and useful tool in many other areas of application, such as social science and economics. They are also used extensively in business and industry in making appropriate decisions in research, development, and production.

Here, in the case of matrices, we have a mathematical method that was developed only for its usefulness in mathematics, but has been found later to be very useful in many areas of application.

Prior to the development of matrices, other methods of solving systems of linear equations were used, but they often involved a great deal of calculation. This led the German mathematician Karl Friedrich Gauss in the early 1800s to develop a systematic method of solving systems of equations, now known as Gaussian elimination. This led to a matrix method that is now used in programming computers to solve systems of equations.

In the final two sections of this chapter, we develop the method of Gaussian elimination, and show the methods that were used to evaluate determinants before the extensive use of calculators and computers.