What you'll learn
This revision guide covers all CSEC-testable content on matrices and transformations. You'll master matrix arithmetic, calculating determinants and inverses, and applying matrices to solve simultaneous equations and describe geometric transformations in the plane. These topics typically account for 10-15% of Paper 2 marks.
Key terms and definitions
Matrix — A rectangular array of numbers arranged in rows and columns, written inside brackets or parentheses.
Order of a matrix — The dimensions of a matrix, written as rows × columns (e.g., a 2×3 matrix has 2 rows and 3 columns).
Determinant — A scalar value calculated from a square matrix that determines whether the matrix has an inverse; for a 2×2 matrix with elements a, b, c, d, the determinant is ad - bc.
Inverse matrix — The matrix A⁻¹ which, when multiplied by the original matrix A, gives the identity matrix (AA⁻¹ = A⁻¹A = I).
Identity matrix — A square matrix with 1s on the main diagonal and 0s elsewhere, denoted I; multiplying any matrix by I leaves it unchanged.
Singular matrix — A square matrix with determinant equal to zero; it has no inverse.
Transformation — A function that maps points in a plane to other points; represented by multiplying position vectors by a transformation matrix.
Invariant point — A point that remains fixed under a transformation; it maps onto itself.
Core concepts
Matrix operations
Addition and subtraction
Matrices must have the same order to be added or subtracted. Add or subtract corresponding elements:
If A = (2 3) and B = (1 -2) (1 4) (3 5)
Then A + B = (3 1) (4 9)
Scalar multiplication
Multiply every element in the matrix by the scalar:
3(2 -1) = (6 -3) (4 0) (12 0)
Matrix multiplication
For matrices A (m×n) and B (n×p), the product AB is defined and has order m×p. Multiply each row of A by each column of B:
(2 1)(3 -1) = (2×3 + 1×2 2×(-1) + 1×4) = (8 2) (0 3)(2 4) (0×3 + 3×2 0×(-1) + 3×4) (6 12)
Important: Matrix multiplication is not commutative; AB ≠ BA in general.
Determinants and inverses of 2×2 matrices
For matrix A = (a b) (c d)
Determinant: det(A) = |A| = ad - bc
Inverse: A⁻¹ = 1/(ad-bc) ( d -b) (-c a)
The inverse exists only when det(A) ≠ 0.
Properties of determinants
- If |A| = 0, the matrix is singular and has no inverse
- If |A| ≠ 0, the matrix is non-singular and has an inverse
- For the inverse: AA⁻¹ = A⁻¹A = I = (1 0) (0 1)
Solving simultaneous equations using matrices
Two simultaneous equations can be written in matrix form:
2x + 3y = 8 x - y = -1
Becomes: (2 3)(x) = ( 8) (1 -1)(y) (-1)
Or AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.
Solution method: X = A⁻¹B
- Find det(A)
- Calculate A⁻¹
- Multiply A⁻¹ by B to find X
Transformations in the plane
A transformation matrix T operates on position vectors to map points (x, y) to new points (x', y'):
(x') = T(x) (y') (y)
Standard transformations
Reflection in the x-axis: ( 1 0) ( 0 -1)
Reflection in the y-axis: (-1 0) ( 0 1)
Reflection in the line y = x: (0 1) (1 0)
Reflection in the line y = -x: ( 0 -1) (-1 0)
Rotation about the origin by angle θ anticlockwise: (cos θ -sin θ) (sin θ cos θ)
Common rotations:
- 90° anticlockwise: ( 0 -1) ( 1 0)
- 180°: (-1 0) ( 0 -1)
- 90° clockwise (270° anticlockwise): ( 0 1) (-1 0)
Enlargement, scale factor k, centre O: (k 0) (0 k)
Stretch parallel to x-axis, factor k: (k 0) (0 1)
Stretch parallel to y-axis, factor k: (1 0) (0 k)
Shear parallel to x-axis, factor k: (1 k) (0 1)
Shear parallel to y-axis, factor k: (1 0) (k 1)
Combined transformations
When transformation M is followed by transformation N, the combined transformation is NM (note the order reversal).
The point undergoes transformation M first, then the result undergoes transformation N:
N(M(x)) = (NM)(x) (y) (y)
Finding invariant points and lines
Invariant points satisfy T(x) = (x) (y) (y)
This gives: (a b)(x) = (x) (c d)(y) (y)
Leading to: ax + by = x and cx + dy = y
Rearrange: (a-1)x + by = 0 and cx + (d-1)y = 0
Solve to find invariant point(s). If only the origin is invariant, the matrix represents a rotation or enlargement.
Invariant lines contain points that map onto other points on the same line (not necessarily themselves).
Worked examples
Example 1: Matrix operations and inverse
Given A = (3 2) and B = (1 -1) (1 -1) (2 3)
(a) Find AB and BA. [4 marks]
Solution:
AB = (3 2)(1 -1) = (3×1 + 2×2 3×(-1) + 2×3) = (7 3) (1 -1)(2 3) (1×1 + (-1)×2 1×(-1) + (-1)×3) (-1 -4)
BA = (1 -1)(3 2) = (1×3 + (-1)×1 1×2 + (-1)×(-1)) = (2 3) (2 3)(1 -1) (2×3 + 3×1 2×2 + 3×(-1)) (9 1)
Note: AB ≠ BA ✓
(b) Find A⁻¹. [3 marks]
Solution:
det(A) = 3×(-1) - 2×1 = -3 - 2 = -5 ✓
A⁻¹ = 1/(-5)(-1 -2) = (1/5 2/5) ✓✓ (-1 3) (1/5 -3/5)
Example 2: Solving simultaneous equations
A coconut vendor in Bridgetown sells small and large bags of coconuts. On Monday, 5 small bags and 3 large bags cost $47. On Tuesday, 2 small bags and 4 large bags cost $38.
(a) Set up this information in matrix form. [2 marks]
(b) Use matrices to find the cost of each size bag. [5 marks]
Solution:
(a) Let x = cost of small bag, y = cost of large bag
(5 3)(x) = (47) ✓✓ (2 4)(y) (38)
(b) Let A = (5 3) (2 4)
det(A) = 5×4 - 3×2 = 20 - 6 = 14 ✓
A⁻¹ = 1/14( 4 -3) = ( 2/7 -3/14) ✓✓ (-2 5) (-1/7 5/14)
(x) = ( 2/7 -3/14)(47) = (2/7 × 47 - 3/14 × 38) = (94/7 - 57/7) = (37/7) (y) (-1/7 5/14)(38) (-1/7 × 47 + 5/14 × 38) (-47/7 + 95/7) (48/7)
x = 37/7 ≈ $5.29, y = 48/7 ≈ $6.86 ✓
Small bag: $5.29, Large bag: $6.86 ✓
Example 3: Transformations
(a) Describe fully the transformation represented by M = ( 0 1). [2 marks] (-1 0)
(b) The point P(4, 2) is transformed by M to point P'. Find the coordinates of P'. [2 marks]
(c) Find the invariant points of this transformation. [3 marks]
Solution:
(a) M represents a rotation of 90° clockwise about the origin ✓✓ (or 270° anticlockwise)
(b) (x') = ( 0 1)(4) = ( 2) ✓ (y') (-1 0)(2) (-4)
P' is (2, -4) ✓
(c) For invariant points: ( 0 1)(x) = (x) (-1 0)(y) (y)
y = x and -x = y ✓
From first equation: y = x Substitute into second: -x = x, so 2x = 0, x = 0 ✓
Therefore y = 0 ✓
The only invariant point is (0, 0), the origin.
Common mistakes and how to avoid them
Multiplying matrices in the wrong order: Remember matrix multiplication is not commutative. When finding a combined transformation, the first transformation applied goes on the right: if M is followed by N, write NM not MN.
Incorrect determinant calculation: For (a b), the determinant is ad - bc, not ab - cd. Watch the diagonal pattern carefully. (c d)
Sign errors in finding the inverse: When writing the inverse, swap the elements on the main diagonal (a becomes d, d becomes a) and change the signs of b and c. Then multiply by 1/det(A).
Forgetting to check if an inverse exists: Always calculate the determinant first. If det(A) = 0, state that the matrix is singular and has no inverse — don't attempt to divide by zero.
Misidentifying transformations: Learn the standard transformation matrices thoroughly. A rotation by 90° anticlockwise is ( 0 -1), not (-1 0). Check your angle and direction carefully. ( 1 0) ( 0 -1)
Confusing invariant points with invariant lines: An invariant point maps to itself exactly. An invariant line contains points that map to other points on the same line. The origin is invariant under all linear transformations through the origin.
Exam technique for "Matrices and Transformations"
"Find the inverse" requires you to show det(A), state the formula, and perform the calculation. Usually 3 marks: 1 for determinant, 2 for correct inverse. Always simplify fractions.
"Describe fully the transformation" means state the type (reflection/rotation/enlargement/shear) AND give complete geometric detail (line of reflection, centre and angle of rotation, scale factor, etc.). Typically 2-3 marks.
Matrix equation problems award marks for method even if arithmetic errors occur. Show det(A), write A⁻¹ symbolically, then calculate. Keep fractions until the final answer.
Combined transformations questions often ask you to find a single matrix representing two successive transformations. Remember to multiply in reverse order and show your matrix multiplication working for method marks.
Quick revision summary
Matrices are rectangular arrays with operations including addition (same order), multiplication (rows × columns), and finding determinants and inverses for 2×2 matrices. The inverse A⁻¹ exists when det(A) ≠ 0 and is found using A⁻¹ = (1/det(A))×adjugate matrix. Matrices solve simultaneous equations via X = A⁻¹B and represent geometric transformations (reflections, rotations, enlargements, shears). Combined transformations multiply in reverse order: N followed by M gives matrix NM. Invariant points satisfy T(x,y) = (x,y) and for most transformations, only the origin is invariant.