Further Mathematics · Lesson 9

Matrices, determinants & linear transformations

WAECNECO

What the syllabus asks

WAEC: matrices (order, equality, addition, scalar and matrix multiplication, transpose, identity), determinants of 2×2 and 3×3 matrices, inverse of a 2×2 matrix (and 3×3 by cofactors), solution of simultaneous linear equations (inverse method and Cramer's rule), linear transformations of the plane (reflection, rotation, enlargement, shear), composition and inverse transformations

NECO: matrices (order, equality, addition, scalar and matrix multiplication, transpose, identity), determinants of 2×2 and 3×3 matrices, inverse of a 2×2 matrix (and 3×3 by cofactors), solution of simultaneous linear equations (inverse method and Cramer's rule), linear transformations of the plane (reflection, rotation, enlargement, shear), composition and inverse transformations

What you will learn

State the order of a matrix and use equality of matrices to find unknowns.
Add, subtract and multiply matrices, and know when a product is defined.
Evaluate determinants of 2 × 2 and 3 × 3 matrices.
Find the inverse of a 2 × 2 matrix and recognise singular matrices.
Solve two and three simultaneous linear equations using matrices and Cramer's rule.
Find matrices for reflections, rotations, enlargements and shears, and the images of points.
Combine transformations, find inverse transformations and use the determinant as an area scale factor.

Key points

Order = rows × columns; equal matrices match every entry.
Add/subtract same-order matrices entry by entry; scalars multiply every entry; Aᵀ swaps rows and columns.
AB needs columns of A = rows of B; (m × n)(n × p) = m × p; AB ≠ BA in general.
det (a b / c d) = ad − bc; det(AB) = det A · det B.
A⁻¹ = (1/det)(d −b / −c a); det = 0 means singular; (AB)⁻¹ = B⁻¹A⁻¹.
AX = B gives X = A⁻¹B; Cramer: x = Dₓ/D, y = D_y/D (and z = D_z/D).
3 × 3 determinant: expand along a row with signs + − +.
Columns of a transformation matrix are the images of (1, 0) and (0, 1).
"P then Q" = QP; |det M| is the area scale factor; reflections det −1, rotations det 1.
Use M⁻¹ to find an object from its image.

Start here: What a matrix is

A matrix is a rectangular arrangement of numbers in rows and columns, written inside brackets. Matrices are a compact way to store information. A provisions shop in Ibadan might record sales like this:

	Rice (bags)	Beans (bags)	Garri (bags)
Monday	4	2	7
Tuesday	3	5	6

As a matrix: S = (4 2 7 / 3 5 6). In this lesson we write a matrix row by row, with "/" separating rows. So (4 2 7 / 3 5 6) means first row 4, 2, 7 and second row 3, 5, 6.

Order: a matrix with m rows and n columns has order m × n (read "m by n"). S has order 2 × 3. The entry in row i, column j is written a_ij; for S, s₁₃ = 7.

Special matrices: a row matrix (1 × n), a column matrix (m × 1), a square matrix (n × n), the zero matrix O, and the identity matrix I = (1 0 / 0 1) for 2 × 2.

Equality

Two matrices are equal only if they have the same order and every corresponding entry is equal. This turns one matrix equation into several ordinary equations.

Worked example 9.1

Given (x + y 2 / 3 x − y) = (7 2 / 3 1), find x and y.

Equate corresponding entries: x + y = 7 and x − y = 1. We can do this because equal matrices agree entry by entry.
Adding: 2x = 8, so x = 4. Then y = 7 − 4 = 3.
Check: x − y = 1 ✓.

x = 4, y = 3

⚠️ Common mistake: writing the order as columns × rows. It is always rows first.

Try it yourself. State the order of (1 0 −2 5) and of (3 / 8 / 1).

Show solution

1 × 4 (row matrix) and 3 × 1 (column matrix).

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Practice questions

1. Find x and y if (2x y / 4 1) = (6 −5 / 4 1).

Show worked answer

x = 3, y = −5.

2. Given A = (3 −2 / 1 4), find det A and A⁻¹.

Show worked answer

det = 12 + 2 = 14; A⁻¹ = (1/14)(4 2 / −1 3).

3. Find (1 −1 / 2 3)(4 / 2).

Show worked answer

(4 − 2 / 8 + 6) = (2 / 14).

4. Solve 3x − 2y = 4, x + y = 3 by Cramer's rule.

Show worked answer

D = 5; Dₓ = 4 + 6 = 10; D_y = 9 − 4 = 5; x = 2, y = 1.

5. Evaluate |2 1 0 / 1 3 1 / 0 1 2|.

Show worked answer

2(6 − 1) − 1(2 − 0) + 0 = 8.

Key words

Matrix: A rectangular array of numbers in rows and columns.
Order: The size of a matrix, rows × columns, e.g. 2 × 3.
Square matrix: A matrix with the same number of rows and columns.
Identity matrix I: A square matrix with 1s on the leading diagonal and 0s elsewhere; AI = IA = A.
Zero (null) matrix: A matrix with every entry 0.
Transpose Aᵀ: The matrix formed by turning rows of A into columns.
Determinant: A number calculated from a square matrix; for 2×2, ad − bc.
Singular matrix: A square matrix whose determinant is 0; it has no inverse.
Inverse A⁻¹: The matrix with AA⁻¹ = A⁻¹A = I.
Minor: The determinant left when a row and a column are deleted.
Cofactor: A minor with the sign (+ or −) given by its position.
Cramer's rule: Solving linear equations by ratios of determinants.

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