Matrices · Lesson 1 of 8
Representing Information with Matrices
“Learn to organise information, interpret entries, and build matrices from rules.”
• Represent labelled information as an ordered rectangular array. • Identify rows, columns, order, and individual entries using aᵢⱼ notation. • Find possible matrix orders from the number of entries. • Construct a matrix from an entry rule and interpret coordinate arrays. • Explain why the meanings of row and column positions must be kept consistent.
From information to an array
A collection of numbers becomes useful when we know what each number describes. A matrix keeps related information in a fixed arrangement so that positions carry meaning. Before calculating with matrices, we will learn to read that arrangement accurately.
Suppose Radha owns 15 notebooks and 6 pens. The row [15, 6] records both quantities, provided we agree that notebooks come first. If we reverse the order of the quantities without changing that agreement, the same two numbers communicate different information. A matrix is therefore ordered: the location of an entry matters as much as its value.
| Person | Notebooks | Pens |
|---|---|---|
| Radha | 15 | 6 |
| Fauzia | 10 | 2 |
| Simran | 13 | 5 |
Removing the descriptive labels leaves three rows of two numbers. The labels remain part of our interpretation even when they are written outside the brackets. Row 1 describes Radha, row 2 Fauzia, and row 3 Simran. Column 1 describes notebooks and column 2 pens. Numbers on a horizontal line form a row; numbers on a vertical line form a column.
A matrix is an ordered rectangular array of numbers or functions. Each individual number or function in the array is an element, also called an entry. Rectangular means that every row has the same number of entries.
A matrix may have just one row, just one column, or even one entry. Thus [15] is already a matrix. In this chapter we work with real numbers and functions whose values are real wherever they are defined. An entry such as sin x is allowed; the definition is not restricted to whole numbers. An irregular arrangement with two entries in one row and three in another is not one matrix.
Matrices give a compact language for systems of linear equations and for information in spreadsheets. The same basic idea supports budgets, sales forecasts, experiment records, and cost comparisons. Matrices can also represent geometric transformations such as rotations and reflections, and have uses in cryptography and other fields. Those applications depend on the elementary ideas we begin here; we do not need their advanced machinery to understand a matrix.
Order and the address of an entry
To read an entry unambiguously, we need the size of the array and an address inside it. The size tells us which row and column numbers are possible. The address identifies a single intersection, just as a row and seat number identify one place in a hall.
A matrix with m rows and n columns has order m × n, read as “m by n”. Rows are counted first. Its total number of entries is mn.
Our possessions matrix has order 3 × 2 and contains 6 entries. Its a₂₁ is 10 because row 2 describes Fauzia and column 1 describes notebooks. Its a₁₂ is 6. Reversing the subscripts usually changes the entry: a₂₁ and a₁₂ refer to different positions. If A has order 3 × 2, a₂₃ does not exist because there is no third column.
For a general matrix A, the ith row is aᵢ₁, aᵢ₂, …, aᵢₙ: the row number remains fixed and the column number changes. The jth column is a₁ⱼ, a₂ⱼ, …, aₘⱼ: the column number remains fixed and the row number changes. These patterns help us translate between a full array and the shorter notation [aᵢⱼ].
In compact matrices written in a sentence, a semicolon separates rows. For instance, [1, 2; 3, 4] means a first row containing 1 and 2 and a second row containing 3 and 4. A semicolon is a layout aid, not an entry.
Problem
Three factories have men/women worker counts (30, 25), (25, 31), and (27, 26), respectively. Represent the information as a 3 × 2 matrix and interpret its third-row, second-column entry.
- 1.Use one row per factory and keep the column order fixed as men, women. The matrix is W = [30, 25; 25, 31; 27, 26].
- 2.There are three rows because there are three factories, and two columns because two worker categories are being recorded.
- 3.The entry w₃₂ is 26. Row 3 selects factory III, and column 2 selects women workers. Thus factory III has 26 women workers.
- 4.The entry 25 occurs twice but has different meanings in its two positions. Repeated values do not make their positions interchangeable.
Order 3 × 2 means three rows and two columns. It does not mean “either three or two”. The matrices of orders 3 × 2 and 2 × 3 contain the same number of entries but have different shapes.
Different arrangements can describe the same data
The categories in a table can be assigned to rows or to columns. Both choices can be useful, but they produce different matrices. We must change the interpretation of every position consistently when changing the arrangement.
The two arrays contain the same recorded facts, but they are not the same matrix. Later we will give this systematic exchange of rows and columns a name. For now, the central point is that changing the arrangement changes how we locate an entry. B’s second-row, third-column entry is 5, the number of pens Simran owns.
Coordinates provide another example. A point P(x, y) can be recorded as the row [x, y] or the column [x; y]. Either representation must preserve the agreement about which position describes x and which describes y. Several points can be placed side by side as columns, or one below another as rows.
In X, the first row contains all four x-coordinates and the second row all four y-coordinates. In Y, each row contains both coordinates of one vertex. These arrays encode the vertices; they do not by themselves draw the edges. To describe the quadrilateral we also retain the intended vertex order A, B, C, D.
Finding sizes and generating entries
Sometimes the array is not given explicitly. We may know only its number of entries, or a rule for calculating an entry from its indices. The definition of order and the meaning of aᵢⱼ give a direct method in both situations.
Problem
A matrix contains exactly 8 entries. Find all its possible orders.
- 1.If the order is m × n, then mn = 8, with m and n positive integers. Find factor pairs rather than guessing an arrangement.
- 2.The ordered pairs are (1,8), (2,4), (4,2), and (8,1). They give orders 1 × 8, 2 × 4, 4 × 2, and 8 × 1.
- 3.Both 2 × 4 and 4 × 2 must be included. They use the same factors but designate different numbers of rows.
- 4.An order such as 0 × 8 is not allowed here, and 3 × 3 has 9 entries rather than 8.
Problem
Construct the 3 × 2 matrix with aᵢⱼ = ½|i − 3j|.
- 1.The allowed row numbers are i = 1, 2, 3 and the allowed column numbers are j = 1, 2. Calculate one entry for each of these six ordered pairs.
- 2.For column 1, substitute j = 1: a₁₁ = ½|1−3| = 1, a₂₁ = ½|2−3| = ½, and a₃₁ = ½|3−3| = 0.
- 3.For column 2, substitute j = 2: a₁₂ = ½|1−6| = 5/2, a₂₂ = ½|2−6| = 2, and a₃₂ = ½|3−6| = 3/2.
- 4.Put each result at its calculated address. The matrix is A = [1, 5/2; 1/2, 2; 0, 3/2].
- 5.Check both shape and rule: there are three rows, two columns, and the entry in position (3,2) is indeed 3/2. The indices are inputs to the rule, not quantities to write as entries unchanged.
Quiz
A matrix has 4 rows and 3 columns. What is its order?
Rows of A represent people and columns represent books, pens, and pencils in that order. What does a₂₃ describe?
Which order is impossible for a matrix with 12 entries?
For aᵢⱼ = 2i − j, what is a₃₂?
Which entry cannot occur in a 2 × 4 matrix?
A 2 × 5 coordinate matrix records one point per column. How many points are recorded?
Practice Problems
- For A = [2, 5, 19, −7; 35, −2, 5/2, 12; √3, 1, −5, 17], state its order and number of entries. Read a₁₃, a₂₁, a₃₃, a₂₄, and a₂₃, explaining how you locate each.
- List every possible order of a matrix with 24 entries, and then with 13 entries. Explain why the prime number 13 gives fewer possible orders.
- List every possible order with 18 entries, and then with 5 entries. Treat reversed factor pairs as different orders.
- Construct three separate 2 × 2 matrices using aᵢⱼ = (i+j)²/2, aᵢⱼ = i/j, and aᵢⱼ = (i+2j)²/2. Write the calculation for every entry.
- Construct two 3 × 4 matrices using aᵢⱼ = ½|−3i+j| and aᵢⱼ = 2i−j. Verify an entry from the last row in each matrix.
- Record points P(−2,3), Q(0,1), and R(4,−1) first as columns and then as rows. State both orders and identify the entry representing Q’s y-coordinate.
- Make a table of the notebooks and pens owned by three imaginary people, then convert it into a matrix in both orientations. Explain the meaning of position (2,1) in each arrangement.
Key Takeaways
• A matrix is an ordered rectangular array; row and column labels give its entries meaning. • Order m × n records rows first and columns second, giving mn entries. • The address aᵢⱼ selects row i and column j within the matrix’s bounds. • Possible orders come from ordered positive factor pairs of the entry count. • An entry rule must be evaluated at every allowed index pair; data and coordinates can be arranged in different consistent orientations.
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Matrix Types and Equality