3.1 Introduction to matrices

A matrix is a rectangular arrangement of numbers in rows and columns, enclosed in brackets and treated as a single object. Matrices are a compact way of holding related business data, such as the quantities of several items ordered from several suppliers, or the prices each supplier quotes for each item, so that whole sets of figures can be added, compared and multiplied in one operation. Example: a procurement officer comparing three suppliers for two stationery items can hold all six quoted prices in one 3 × 2 matrix, one row per supplier and one column per item.

Notation and elements

A matrix is named with a capital letter, and its numbers, called elements or entries, are written inside square or round brackets. In this text matrices are set out row by row as below.

A = | 2   5   1 |
    | 4   0   3 |

• Rows: The horizontal lines of elements. Matrix A has 2 rows: (2, 5, 1) and (4, 0, 3).
• Columns: The vertical lines of elements. Matrix A has 3 columns: (2, 4), (5, 0) and (1, 3).
• Element position: The element in row i and column j is written aij. In matrix A, a12 = 5 (row 1, column 2) and a23 = 3 (row 2, column 3).
• Order (size): The number of rows by the number of columns, written m × n and read "m by n". Matrix A is of order 2 × 3; it has 2 × 3 = 6 elements.
• Rows come first: A 2 × 3 matrix and a 3 × 2 matrix are different shapes, so the order is always stated as rows × columns.

Example: a business data matrix

A hardware store buys three items from two suppliers in one month. The quantities, in units, are:

As a matrix, with suppliers as rows and items as columns:

Q = | 120   40   25 |
    |  80   60   30 |

Q is a 2 × 3 matrix. The element q21 = 80 is the number of cement bags bought from Supplier Q. The labels are kept in the table or in the description; the matrix holds only the numbers, in a fixed order that everyone using it must know.

Types of matrices

• Row matrix: Has only one row, for example (5 8 2), of order 1 × 3. It can hold the prices of three items from one supplier.
• Column matrix: Has only one column, of order n × 1, for example the quantities of three items written one under the other.
• Square matrix: Has the same number of rows and columns (n × n), for example 2 × 2. Determinants and inverses (3.4 and 3.5) exist only for square matrices.
• Zero (null) matrix: Every element is 0. Written O, it behaves like the number 0 in addition.
• Diagonal matrix: A square matrix whose elements off the leading diagonal (top left to bottom right) are all 0.
• Identity (unit) matrix: A diagonal matrix whose leading diagonal elements are all 1. Written I, it behaves like the number 1 in multiplication.
• Scalar matrix: A diagonal matrix whose diagonal elements are all equal, for example 3 on the diagonal and 0 elsewhere.
• Symmetric matrix: A square matrix that is unchanged when its rows and columns are interchanged, so that aij = aji.

The 2 × 2 identity and zero matrices are:

I = | 1   0 |        O = | 0   0 |
    | 0   1 |            | 0   0 |

Equality of matrices

Two matrices are equal only when they have the same order and every pair of corresponding elements is equal. This is used to find unknown elements.

Example: if the matrices below are equal, find x and y.

| x + 2    6 |   =   | 7     6 |
| 3      2y  |       | 3    10 |

Comparing corresponding elements:

x + 2 = 7, so x = 5

2y = 10, so y = 5

Transpose of a matrix

The transpose of a matrix A, written A' or Aᵀ, is formed by writing its rows as columns. A matrix of order m × n has a transpose of order n × m. Transposing is useful when data has to be rearranged so that two matrices fit together for multiplication (3.2.4).

A  = | 2   5   1 |          Aᵀ = | 2   4 |
     | 4   0   3 |               | 5   0 |
                                 | 1   3 |

Uses of matrices in procurement and business

• Organising data: Quantities, prices and costs for several items and suppliers are held in one structure.
• Calculating totals and costs: Multiplying a quantity matrix by a price matrix gives the total cost from each supplier in one step (3.2.4 and 3.6).
• Comparing suppliers or periods: Subtracting one month's matrix from another shows the change for every item and supplier at once.
• Solving simultaneous equations: Two related unknowns, such as the unit prices of two items, can be found with the inverse of a 2 × 2 matrix (3.3 to 3.5).
• Spreadsheets and software: Spreadsheet programs store data in rows and columns and have functions for matrix multiplication and inverses, so matrix thinking carries directly into everyday computer work.

Chapter Summary

This chapter introduced the fundamental concept of matrices, emphasizing their structure and significance in various calculations. It covered essential matrix operations including addition, subtraction, division, and multiplication, detailing how these operations are performed and their mathematical rules. The chapter then focused on solving 2 by 2 matrix equations, providing methods to handle such systems effectively. Determinants of 2 by 2 matrices were explained as a critical value for matrix properties and solution existence. The concept of the inverse of a 2 by 2 matrix was explored, showing how to compute it and why it is important for solving linear equations. Finally, the chapter demonstrated practical applications of matrices in solving business operations, highlighting their usefulness in real-world problem solving and decision-making processes.