What is a matrix?

A matrix is a rectangular table of numbers in rows (across) and columns (down). It is a neat way to store and work with many numbers at once: marks of several students in several subjects, prices of several items in several shops.

Its order is written m × n: m rows, then n columns, always in that order. An m × n matrix has m × n entries. The entry in row i and column j is called aᵢⱼ, so a₂₃ sits in row 2, column 3.

In this lesson we write a matrix row by row, with a semicolon between rows: [2, 5; 1, 0] has first row 2, 5 and second row 1, 0. It is a 2 × 2 matrix.

Building a matrix from a rule. If aᵢⱼ = i + 2j for a 2 × 2 matrix, then a₁₁ = 3, a₁₂ = 5, a₂₁ = 4, a₂₂ = 6, so the matrix is [3, 5; 4, 6].

Kinds of matrix, by shape and contents:

• Row matrix: one row. Column matrix: one column.
• Square matrix: as many rows as columns (m = n). Its diagonal runs from top-left to bottom-right.
• Diagonal matrix: a square matrix with zeros everywhere off the diagonal.
• Scalar matrix: a diagonal matrix whose diagonal entries are all the same number.
• Identity matrix I: a scalar matrix with 1s on the diagonal. It plays the role that 1 plays among numbers.
• Zero matrix O: every entry is 0.

Equal matrices. Two matrices are equal only when they have the same order and every matching entry is equal. So one matrix equation gives you several ordinary equations, one per position.

Example: [x + y, 2; 5, x – y] = [7, 2; 5, 1] gives x + y = 7 and x – y = 1, so x = 4 and y = 3.

In an answer, write one line such as 'on comparing the corresponding entries, we get' before listing the equations, and number them. Examiners look for that step.