A complete guide to matrix arithmetic, determinants, inverses, solving simultaneous equations, and transformation matrices for reflection and rotation.
A matrix is a rectangular grid of numbers, arranged in rows and columns, used to organise data and perform calculations efficiently. Matrices have their own arithmetic — addition, subtraction, multiplication, and special quantities like the determinant and inverse — and one of their most striking applications is representing the geometric transformations covered earlier in this subcategory as a single multiplication. This guide builds matrix arithmetic from the ground up and connects it directly to reflection and rotation.
A matrix is written as a grid of numbers inside brackets, e.g. (1 2 over 3 4). Each number is called an element, and its position is described by its row and column.
The order of a matrix is given as (number of rows) × (number of columns). A matrix with 2 rows and 3 columns has order 2 × 3. This course focuses mainly on 2 × 2 matrices, the most common size in O Level work.
Matrices can only be added or subtracted if they have the same order. Corresponding elements (same row and column position) are added or subtracted directly.
A = (1 2 over 3 4), B = (5 0 over −1 2). Find A + B.
A + B = (1+5 2+0 over 3+(−1) 4+2) = (6 2 over 2 6).
Multiplying a matrix by a scalar (a plain number) multiplies every element by that number.
A = (2 −3 over 1 4). Find 3A.
3A = (6 −9 over 3 12).
Matrix multiplication is not done element-by-element. To multiply two matrices, each element of the result is found by taking a row from the first matrix and a column from the second, multiplying corresponding entries, and adding the products ("row times column").
A = (1 2 over 3 4), B = (5 6 over 7 8). Find AB.
Top-left: (1×5)+(2×7) = 19. Top-right: (1×6)+(2×8) = 22. Bottom-left: (3×5)+(4×7) = 43. Bottom-right: (3×6)+(4×8) = 50.
AB = (19 22 over 43 50).
Matrix multiplication is generally not commutative: AB usually does not equal BA. Always keep the order exactly as given in a problem.
The identity matrix, I = (1 0 over 0 1), behaves like the number 1 in ordinary multiplication: multiplying any matrix by I (in either order) leaves it unchanged.
A = (4 3 over 2 5). Find det(A).
det(A) = (4×5) − (3×2) = 20 − 6 = 14.
The inverse of matrix A, written A⁻¹, satisfies AA⁻¹ = I. For a 2×2 matrix A = (a b over c d), the inverse is found by swapping a and d, negating b and c, and dividing every element by the determinant: A⁻¹ = 1/det(A) × (d −b over −c a). A matrix only has an inverse if its determinant is not zero (a matrix with determinant 0 is called singular).
A = (4 3 over 2 5), with det(A) = 14 (from Example 4). Find A⁻¹.
A⁻¹ = 1/14 × (5 −3 over −2 4) = (5/14 −3/14 over −2/14 4/14).
A pair of simultaneous equations can be written as a single matrix equation, AX = B, and solved using the inverse: X = A⁻¹B.
Solve: 2x + y = 8, x + 3y = 9, using matrices.
In matrix form: (2 1 over 1 3)(x over y) = (8 over 9). det = (2×3)−(1×1) = 5.
Inverse = 1/5 × (3 −1 over −1 2). So (x over y) = 1/5 × (3 −1 over −1 2)(8 over 9) = 1/5 × (24−9 over −8+18) = 1/5 × (15 over 10) = (3 over 2). So x = 3, y = 2.
A 2×2 matrix can represent a geometric transformation: multiplying the matrix by a point's column vector (x over y) gives the image point's coordinates. This connects directly to the four transformations covered earlier in this subcategory.
| Transformation | Matrix |
|---|---|
| Reflection in the x-axis | (1 0 over 0 −1) |
| Reflection in the y-axis | (−1 0 over 0 1) |
| Reflection in y = x | (0 1 over 1 0) |
| Rotation 90° anticlockwise about the origin | (0 −1 over 1 0) |
| Rotation 180° about the origin | (−1 0 over 0 −1) |
Use the reflection-in-y=x matrix to reflect point (5, 2).
(0 1 over 1 0)(5 over 2) = ((0×5)+(1×2) over (1×5)+(0×2)) = (2 over 5) — matching the "swap coordinates" rule from the Transformations article.
Applying two transformations in sequence corresponds to multiplying their matrices together — but the order matters, since matrix multiplication is not commutative. To apply transformation P first, then Q, the combined matrix is QP (Q multiplied on the left of P), since the point vector is multiplied on the right.
Common errors include: adding or multiplying matrices of incompatible orders; trying to multiply matrices element-by-element instead of using the row-times-column method; forgetting that matrix multiplication order matters (AB ≠ BA in general); and forgetting to divide by the determinant when finding an inverse.
| Operation | Rule |
|---|---|
| Addition/subtraction | Requires the same order; combine corresponding elements |
| Scalar multiplication | Multiply every element |
| Matrix multiplication | Row times column, order matters |
| Determinant (2×2) | ad − bc |
| Inverse (2×2) | 1/det(A) × (d −b over −c a) |
This article is original EDUSAMBAM educational writing. It is designed as a broad matrices resource covering arithmetic, determinants, inverses and transformation applications for confident problem-solving. Exact examination requirements can vary between examination boards and syllabuses, so students should also compare their work with the specification and past-paper requirements of their own board.
Recommended study approach: practise matrix addition, scalar multiplication and row-times-column multiplication until automatic, then learn the determinant and inverse formulas before applying them to simultaneous equations and transformation matrices.
20 questions covering matrix arithmetic, determinants, inverses, simultaneous equations, and transformation matrices. Answer every question, then submit to see your score instantly.