Mathematics · Lesson 13

Matrices & determinants

JAMBWAECNECO

What the syllabus asks

JAMB: Matrices and determinants: algebra of matrices not exceeding 3 × 3, determinants of matrices not exceeding 3 × 3, inverses of 2 × 2 matrices (excluding quadratic and higher-degree equations)

WAEC: Matrices and determinants: order and notation, equality, addition, subtraction, scalar multiplication, multiplication of matrices (up to 3 × 3), identity matrix, determinant of a 2 × 2 matrix, inverse of a 2 × 2 matrix, solving simultaneous linear equations by the matrix method

NECO: Matrices and determinants: order and notation, equality, addition, subtraction, scalar multiplication, multiplication of matrices (up to 3 × 3), identity matrix, determinant of a 2 × 2 matrix, inverse of a 2 × 2 matrix, solving simultaneous linear equations by the matrix method

What you will learn

State the order of a matrix and identify row, column, square, zero and identity matrices.
Add, subtract and multiply matrices by a scalar, and use equality of matrices to find unknowns.
Multiply matrices up to 3 × 3, deciding first whether the product exists.
Recognise that matrix multiplication is not commutative and use the identity matrix.
Find the transpose of a matrix.
Calculate determinants of 2 × 2 and 3 × 3 matrices and recognise singular matrices.
Find the inverse of a 2 × 2 matrix and use it to solve a pair of simultaneous linear equations.

Key points

Order = rows × columns; a 2 × 3 matrix has 2 rows and 3 columns.
Equal matrices have the same order and equal matching elements.
Add and subtract only matrices of the same order, element by element.
A scalar multiplies every element.
AB exists when columns of A = rows of B; each entry is a row × a column; AB ≠ BA in general.
AI = IA = A, where I has 1s on the leading diagonal.
The transpose swaps rows and columns; (AB)ᵀ = BᵀAᵀ.
|A| = ad − bc; a singular matrix has determinant 0 and no inverse.
A⁻¹ = (1/|A|)(d −b / −c a); solve AX = B with X = A⁻¹B.
3 × 3 determinants expand along a row with signs + − +; |det| is the area scale factor of a transformation.

Start here: What is a matrix?

Mrs Okafor runs a provisions shop in Onitsha. Every week she records how many cartons of three products she sells on two days. Instead of writing sentences, she writes a neat table of numbers:

	Milk	Sugar	Noodles
Monday	4	2	7
Saturday	9	5	12

If we remove the headings and keep only the numbers in brackets, we have a matrix. A matrix is simply a rectangular arrangement of numbers in rows and columns. Each number is called an element or entry. Matrices let us store and calculate with lots of information at once, which is why they are used in computer graphics, economics, engineering and even in the apps on your phone.

Order of a matrix = number of rows × number of columns (always rows first).

Mrs Okafor’s matrix has 2 rows and 3 columns, so its order is 2 × 3 (read “two by three”). We name matrices with capital letters, such as A or M, and refer to an element by its position: a₁₂ means the element in row 1, column 2. In her matrix, a₁₂ = 2 and a₂₃ = 12.

Special matrices you must recognise:

Row matrix: one row, e.g. (3 −1 5), order 1 × 3.
Column matrix: one column, e.g. two numbers stacked, order 2 × 1.
Square matrix: same number of rows and columns, e.g. 2 × 2 or 3 × 3.
Zero (null) matrix: every element is 0.
Identity matrix I: square, with 1s down the leading diagonal and 0s everywhere else. The 2 × 2 identity is (1 0 / 0 1).

In this lesson we write a 2 × 2 matrix in one line as (a b / c d), where the slash separates the first row from the second. So (2 3 / 1 4) means first row 2, 3 and second row 1, 4.

Worked example 13.1

State the order of each matrix: (a) (2 3 / 1 4); (b) (1 2 0 / 3 −1 4); (c) the column with entries 6, 0, −2.

(a) Two rows and two columns: order 2 × 2. It is a square matrix.
(b) Two rows (separated by the slash) and three numbers in each row: order 2 × 3.
(c) Three rows, one column: order 3 × 1. It is a column matrix.

2 × 2, 2 × 3, 3 × 1

⚠️ Common mistake: writing the order as columns × rows. A 2 × 3 matrix and a 3 × 2 matrix are different shapes. Always say rows first, just as you say “row, column” when finding a seat number in a hall.

Try it yourself. How many elements does a 3 × 4 matrix have? What is its order after you swap its rows and columns?

Show solution

3 × 4 = 12 elements. Swapping rows and columns gives order 4 × 3.

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Practice questions

1. P = (3 −1 / 2 5) and Q = (1 4 / 0 −2). Find P + 2Q.

Show worked answer

2Q = (2 8 / 0 −4); P + 2Q = (5 7 / 2 1).

2. Using the same P and Q, find PQ.

Show worked answer

(3×1 + (−1)×0 3×4 + (−1)(−2) / 2×1 + 5×0 2×4 + 5(−2)) = (3 14 / 2 −2).

3. Find the determinant of P = (3 −1 / 2 5).

Show worked answer

15 − (−2) = 17.

4. Find the inverse of P = (3 −1 / 2 5).

Show worked answer

(1/17)(5 1 / −2 3).

5. Solve 4x + 3y = 10 and 2x + 5y = 12 by the matrix method.

Show worked answer

|A| = 14; X = (1/14)(50 − 36 / −20 + 48) = (1 / 2): x = 1, y = 2.

Key words

Matrix: A rectangular arrangement of numbers in rows and columns, enclosed in brackets.
Element (entry): A single number inside a matrix.
Order: The size of a matrix written as rows × columns.
Square matrix: A matrix with the same number of rows and columns.
Row matrix: A matrix with only one row.
Column matrix: A matrix with only one column.
Zero (null) matrix: A matrix whose elements are all 0.
Identity matrix (I): A square matrix with 1s on the leading diagonal and 0s elsewhere; AI = IA = A.
Leading diagonal: The diagonal from the top-left to the bottom-right of a square matrix.
Scalar: An ordinary number that multiplies a matrix.
Transpose (Aᵀ): The matrix formed by turning rows into columns.
Determinant: A single number calculated from a square matrix; for a 2 × 2 matrix it is ad − bc.

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