MatricesIntroduction

Riya has just photographed the rangoli outside her house for Diwali. She zooms in on her phone, further and further, until the picture breaks up into tiny squares. Each square — a pixel — stores nothing but a number that says how dark it is. So the whole photo is really a rectangle of numbers, arranged in rows and columns. When Riya rotates the photo or makes it darker, her phone is doing arithmetic on that rectangle — on all of it at once.

Here is a tiny 5 × 5 picture of an arrow, with its grid of numbers beside it: 0 is a blank pixel, and bigger numbers are darker. Each button changes all 25 numbers in one go. Complete the three missions.

A rectangle of numbers handled as one single object is called a matrix. One of your buttons swapped the rows of the grid with its columns; the other multiplied every number by 2. By the end of this chapter both buttons will have proper names — the transpose and multiplication by a scalar — and you will be able to add, subtract and multiply whole matrices, and even undo a multiplication.

From Counting Boards to Spreadsheets

The name came much later. In 1850 the English mathematician James Joseph Sylvester called such an array a matrix — Latin for the place where something grows. A few years later his friend Arthur Cayley wrote down rules for adding and multiplying matrices, and turned them into objects you can calculate with, just like numbers.

Today matrices are everywhere. A spreadsheet is a matrix with labels — items down the side, months across the top — and businesses use them for budgets, sales projections and cost estimates. Computer graphics stores the corners of a shape in a matrix, so that magnifying, rotating or reflecting the whole shape is one matrix calculation. Secret codes scramble messages with matrices, and genetics, economics, sociology, psychology and industrial management all depend on them.

In this chapter you will learn the language of matrices — rows, columns and order — meet the important types of matrices, and then learn to add, multiply, transpose and invert them.