Grade 9 Mathematics · Strand 2.0 Algebra · Sub-strand 2.1 Matrices
These are free study notes for Grade 9 learners, written to the KICD curriculum design and taken from our full Grade 9 Mathematics notes. Matrices are new in Grade 9, so learners meet them for the first time here. Once the idea of rows and columns is clear, the calculations are short and reliable marks in an exam.
A football league table, a class timetable and a shop's price list all arrange numbers in rows and columns. Such an arrangement is a matrix. In this topic we identify matrices, state their order, locate their elements, decide when two matrices can be added or subtracted, and use matrix addition and subtraction for real data such as stock in a shop.
What a matrix is
Matrix
A rectangular arrangement of numbers in rows (across) and columns (down), enclosed in brackets. Each number is an element.
In these notes a matrix is written row by row with a semicolon between rows: [5 3 1 ; 5 2 2] has first row 5 3 1 and second row 5 2 2.
From a league table to a matrix
Two teams: Team A played 5, won 3, drew 1; Team B played 5, won 2, drew 2.
Rows = teams, columns = Played, Won, Drawn.
Matrix: [5 3 1 ; 5 2 2].
Order and position of elements
Order of a matrix
The number of rows by the number of columns, written rows × columns. [5 3 1 ; 5 2 2] has order 2 × 3 and 2 × 3 = 6 elements.
| Type | Description | Example and order |
|---|---|---|
| Row matrix | One row | [4 7 9], order 1 × 3 |
| Column matrix | One column | [2 ; 8], order 2 × 1 |
| Square matrix | Same number of rows and columns | [1 0 ; 0 1], order 2 × 2 |
| Rectangular matrix | Different numbers of rows and columns | [5 3 1 ; 5 2 2], order 2 × 3 |
The element in row i and column j is written aij. In A = [4 9 2 ; 7 5 8], a12 = 9 (row 1, column 2) and a23 = 8 (row 2, column 3). Rows are counted from the top and columns from the left.
Adding and subtracting matrices
Two matrices are compatible for addition or subtraction only if they have the same order, so every element has a partner in the same position. Add or subtract the elements in matching positions; the answer has the same order.
Adding and subtracting
[2 1 ; 3 4] + [5 0 ; 1 2] = [2+5 1+0 ; 3+1 4+2] = [7 1 ; 4 6].
[8 3 ; 6 5] - [2 7 ; 6 1] = [6 -4 ; 0 4]. A negative element is allowed.
[1 2] + [4 3] - [2 6] = [3 -1].
[2 1 ; 3 4] + [1 2 3] is not possible: the orders 2 × 2 and 1 × 3 differ.
Matrices for real data
Stock in a school canteen
Rows: buns, sodas. Columns: small, large.
Start of the week: [60 40 ; 48 24]. Deliveries: [30 20 ; 24 12]. Sales: [70 35 ; 50 30].
End stock = start + deliveries - sales.
= [60+30-70 40+20-35 ; 48+24-50 24+12-30] = [20 25 ; 22 6].
League tables, timetables, mark sheets and computer spreadsheets are matrices. A digital photo is stored as a huge matrix with one number for each tiny dot (pixel).
Key points to remember
- A matrix is numbers arranged in rows and columns inside brackets.
- Order = rows × columns; a 2 × 3 matrix has 6 elements.
- aij is the element in row i, column j.
- Only matrices of the same order can be added or subtracted, element by element.
Test yourself: multiple choice
What is the order of [3 0 7 2 ; 1 4 5 6]?
Show the answer
B: It has 2 rows and 4 columns.
In A = [6 1 ; 9 4 ; 3 8], what is a31?
Show the answer
A: Row 3, column 1 is 3.
Which matrix can be added to [2 5 ; 1 3]?
Show the answer
C: Only [3 1 ; 0 2] has the same order, 2 × 2.
What is [7 2 ; 5 9] - [3 4 ; 5 1]?
Show the answer
D: 7-3 = 4, 2-4 = -2, 5-5 = 0, 9-1 = 8.
How many elements does a 3 × 4 matrix have?
Show the answer
B: 3 × 4 = 12.
Practice questions
State the order of each matrix: (a) [1 2 3] (b) [4 ; 5] (c) [1 0 2 ; 3 4 5 ; 6 7 8]. (3 marks)
Show the answer
(a) 1 × 3 (b) 2 × 1 (c) 3 × 3.
Given B = [7 3 0 ; 2 8 5], write down b13 and b22. (2 marks)
Show the answer
b13 = 0; b22 = 8.
Work out [4 -1 ; 0 6] + [-2 5 ; 3 -6]. (2 marks)
Show the answer
[2 4 ; 3 0].
Work out [10 8 6] - [3 8 9] + [1 2 3]. (2 marks)
Show the answer
[10-3+1 8-8+2 6-9+3] = [8 2 0].
A shop had 50 exercise books and 30 pens on Monday, written [50 30]. It received [40 25] and sold [65 32] during the week. Find the stock at the end of the week. (3 marks)
Show the answer
[50+40-65 30+25-32] = [25 23]: 25 books and 23 pens.
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