Matrices

A matrix is a rectangular arrangement of numbers in rows and columns, enclosed in square brackets. Each number is called an element. The size of a matrix is its order, written as 'number of rows x number of columns', with rows always stated first. So a matrix with 2 rows and 3 columns has order 2 x 3 and holds 2 times 3 = 6 elements. An individual element is named a_ij, where i is its row number and j is its column number. Matrices give mathematicians and computers a tidy way to store and handle whole tables of data at once.

Matrices are classified by their shape and contents. A row matrix has a single row; a column matrix has a single column. A square matrix has equal numbers of rows and columns. A diagonal matrix is a square matrix whose non-diagonal elements are all zero. A scalar matrix is a diagonal matrix with all diagonal elements equal. The identity (or unit) matrix is a diagonal matrix whose diagonal elements are all 1. A null (zero) matrix has every element equal to zero. Recognising the type often tells you straight away how it will behave in a calculation.

Two matrices can be added or subtracted only when they have exactly the same order; you simply add or subtract the elements in matching positions. Addition is commutative (A + B = B + A) and associative, and the null matrix of the same order acts as the identity for addition (A + 0 = A). To multiply a matrix by an ordinary number, called a scalar, you multiply every single element by that number. These operations are the gentle, predictable side of matrix algebra.

Multiplying two matrices is the careful part. The product A x B exists only if the number of columns of A equals the number of rows of B; the result then has the rows of A and the columns of B. Each element of the product comes from a row of A and a column of B: multiply the pairs and add them. A crucial fact is that matrix multiplication is not commutative, so AB and BA are usually different and BA may not even exist. The identity matrix I behaves like the number 1: multiplying any suitable matrix by I leaves it unchanged, A I = I A = A.