Matrices: notation and order
A matrix is a rectangular array of numbers in rows and columns. With m rows and n columns it has order m × n and mn entries aᵢⱼ.
What a matrix is
A matrix is a rectangular arrangement of numbers, called elements or entries, written inside brackets. The horizontal lines of numbers are rows and the vertical lines are columns. A matrix with m rows and n columns has order m × n (read 'm by n'). The element in row i and column j is written aᵢⱼ. In this chapter every entry is a real number. A matrix is only an arrangement: it does not have a single numerical value. How we write matrices here: row by row, with rows separated by semicolons. So [2, 5; −1, 3] has first row 2, 5 and second row −1, 3, and [4; 0; −3] is a column.
A matrix as a table: marks of three students
| Student | Maths | Physics |
|---|---|---|
| Anu | 78 | 85 |
| Bala | 92 | 70 |
| Charu | 65 | 88 |
Reading a matrix
- The marks table gives M = [78, 85; 92, 70; 65, 88]: 3 rows (students) and 2 columns (subjects), so M has order 3 × 2 and 6 elements.
- m₂₁ = 92 (row 2, column 1: Bala's Maths mark) and m₃₂ = 88 (Charu's Physics mark).
- Row first, column second, both in the order and in the subscripts. a₂₃ and a₃₂ are different elements.
- A matrix with one row has order 1 × n; a matrix with one column has order m × 1.
- To list all possible orders of a matrix with k elements, write every factor pair of k.
Example 1: build a matrix from a rule
Construct a 2 × 3 matrix A = [aᵢⱼ] with aᵢⱼ = i² − j.
- Rule: put each pair (i, j) into the formula; i runs over the rows 1, 2 and j over the columns 1, 2, 3.
- a₁₁ = 1 − 1 = 0, a₁₂ = 1 − 2 = −1, a₁₃ = 1 − 3 = −2.
- a₂₁ = 4 − 1 = 3, a₂₂ = 4 − 2 = 2, a₂₃ = 4 − 3 = 1.
- Write row 1, then row 2.
A = [0, −1, −2; 3, 2, 1]
Example 2: a rule with an absolute value (board style)
Construct a 3 × 2 matrix A = [aᵢⱼ] with aᵢⱼ = |2i − 3j|/2.
- Rule: i = 1, 2, 3 (rows) and j = 1, 2 (columns).
- Row 1: a₁₁ = |2 − 3|/2 = 1/2, a₁₂ = |2 − 6|/2 = 2.
- Row 2: a₂₁ = |4 − 3|/2 = 1/2, a₂₂ = |4 − 6|/2 = 1.
- Row 3: a₃₁ = |6 − 3|/2 = 3/2, a₃₂ = |6 − 6|/2 = 0.
A = [1/2, 2; 1/2, 1; 3/2, 0]
A school timetable, a sheet of monthly electricity units for each flat in an apartment block, and the pixel grid of a phone photo are all matrices: numbers arranged in rows and columns.
1 marks: “Write all possible orders of a matrix with 10 elements.” The factor pairs of 10: 1 × 10, 2 × 5, 5 × 2 and 10 × 1.
2 marks: “Construct a 2 × 2 matrix A = [aᵢⱼ] with aᵢⱼ = (2i − j)²/3.” a₁₁ = 1/3, a₁₂ = 0, a₂₁ = 9/3 = 3, a₂₂ = 4/3. So A = [1/3, 0; 3, 4/3].
Rows first, then columns, like 'RC' on a railway coach. Both the order m × n and the element aᵢⱼ put the row first.