Matrices and Determinants

By M Bourne

Introduction

Why study Matrices?

A matrix is simply a set of numbers arranged in a rectangular table.

2x4 matrix

Shown is an example of a 2 × 4 matrix. It has 2 rows and 4 columns. We usually write matrices inside parentheses ( ) or brackets [ ].

We can add, subtract and multiply matrices together, under certain conditions.

matrices

We use matrices to solve simultaneous equations, that we met earlier.

Matrices are used to solve problems in:

  • electronics
  • statics
  • robotics
  • linear programming
  • optimisation
  • intersections of planes
  • genetics

We see several of these applications throughout this chapter, especially in Matrices and Linear Equations.

For large systems of equations, we use a computer to find the solution. This chapter first shows you the basics of matrix arithmetic, and then we show some computer examples (using Scientific Notebook and LiveMath) so that you understand what the computer is doing for you.

You can skip over the next part if you want to go straight to matrices.

Determinants

Continue reading below

A determinant of a matrix represents a single number. We obtain this value by multiplying and adding its elements in a special way. We can use the determinant of a matrix to solve a system of simultaneous equations.

For example, if we have the (square) 2 × 2 matrix:

2x2

The determinant of this matrix is written within vertical lines as follows:

2x2 determinant

We'll see in the next section how to evaluate this determinant. (It has value -29).