📚 IGCSE AQA Mathematics: Matrices Revision Notes | IGCSE AQA 数学：矩阵 考点精讲
Matrices are rectangular arrays of numbers that appear throughout IGCSE AQA Mathematics. They provide a powerful way to store data, solve simultaneous equations, and describe transformations such as reflections, rotations and enlargements. Understanding how to handle matrices – from basic arithmetic to finding inverses – is a must for any student aiming for top marks in this topic.
矩阵是由数字排列而成的矩形阵列，在 IGCSE AQA 数学中无处不在。它们为存储数据、求解联立方程、描述反射、旋转和放大等变换提供了有力的工具。扎实掌握矩阵的基本运算与逆矩阵等概念，是每位同学在本专题中取得高分的必备条件。
1. Introduction to Matrices | 矩阵简介
A matrix is described by its order, written as rows × columns. For example, a matrix with 2 rows and 3 columns has order 2 × 3. The entry in row i and column j is denoted by aᵢⱼ.
矩阵由它的 阶（行数 × 列数）来描述。比如一个 2 行 3 列的矩阵，其阶为 2 × 3。位于第 i 行、第 j 列的元素记作 aᵢⱼ。
Two matrices are equal only when they have the same order and each pair of corresponding elements is identical. When working with equal matrices you can set up equations element‑by‑element.
只有当两个矩阵阶数相同且每一对对应元素相等时，这两个矩阵才相等。处理矩阵相等时，可以逐元素建立方程。
2. Matrix Addition and Subtraction | 矩阵的加法与减法
You can add or subtract matrices only if they have exactly the same order. The result is found by adding or subtracting the elements in matching positions.
只有阶数完全相同的矩阵才能相加或相减。结果是通过对应位置的元素相加或相减得到的。
A ± B = [aᵢⱼ ± bᵢⱼ]
A ± B = [aᵢⱼ ± bᵢⱼ]
Example: If A = [1 2; 3 4] and B = [5 6; 7 8], then A + B = [6 8; 10 12] and A – B = [–4 –4; –4 –4].
示例：若 A = [1 2; 3 4]，B = [5 6; 7 8]，则 A + B = [6 8; 10 12]，A – B = [–4 –4; –4 –4]。
3. Scalar Multiplication | 标量乘法
Multiplying a matrix by a scalar (an ordinary number) means multiplying every element of the matrix by that number. The order of the matrix stays the same.
矩阵与标量（一个普通数）相乘，就是将矩阵中的 每一个 元素都乘以该数。矩阵的阶保持不变。
k × [a b; c d] = [ka kb; kc kd]
k × [a b; c d] = [ka kb; kc kd]
Scalar multiplication is used to factor common numbers out of every entry, and it lays the groundwork for transformation enlargements.
标量乘法常用于提取公因子，也为放大变换奠定了基础。
4. Matrix Multiplication | 矩阵乘法
Two matrices can only be multiplied when the number of columns in the first equals the number of rows in the second. If A is m×n and B is n×p, the product AB will be a matrix of order m×p.
两个矩阵只有在 第一个矩阵的列数 等于 第二个矩阵的行数 时才能相乘。若 A 是 m×n 矩阵，B 是 n×p 矩阵，则乘积 AB 是一个 m×p 矩阵。
The element in row i, column j of AB is the dot product of the i‑th row of A with the j‑th column of B.
AB 中第 i 行第 j 列的元素是 A 的第 i 行与 B 的第 j 列的点积。
For example, to find AB where A = [1 2; 3 4] and B = [2 0; 1 3]: the top‑left entry is (1×2 + 2×1) = 4; top‑right is (1×0 + 2×3) = 6; bottom‑left is (3×2 + 4×1) = 10; bottom‑right is (3×0 + 4×3) = 12, so AB = [4 6; 10 12].
例如，计算 AB，其中 A = [1 2; 3 4] 和 B = [2 0; 1 3]：左上角元素为 (1×2 + 2×1) = 4；右上角为 (1×0 + 2×3) = 6；左下角为 (3×2 + 4×1) = 10；右下角为 (3×0 + 4×3) = 12，因此 AB = [4 6; 10 12]。
5. Properties of Matrix Multiplication | 矩阵乘法的性质
Matrix multiplication is not commutative: in general, AB ≠ BA. The example above gives BA = [2 4; 10 18], which differs from AB.
矩阵乘法 不满足交换律：一般情况下 AB ≠ BA。上例中 BA = [2 4; 10 18]，与 AB 不同。
Multiplication is associative: (AB)C = A(BC), and it is distributive over addition: A(B + C) = AB + AC and (A + B)C = AC + BC. These properties allow you to rearrange brackets but never change the order of the matrices.
乘法满足 结合律：(AB)C = A(BC)，并且对加法满足 分配律：A(B + C) = AB + AC 以及 (A + B)C = AC + BC。这些性质允许重新安排括号，但绝不能改变矩阵的顺序。
6. Zero and Identity Matrices | 零矩阵与单位矩阵
The zero matrix O is a matrix where every element is zero. Adding O to any matrix A leaves A unchanged: A + O = O + A = A. Multiplication by a zero matrix of compatible order gives a zero matrix: A × O = O (and O × A = O).
零矩阵 O 是所有元素均为 0 的矩阵。任何矩阵 A 加上 O 保持原样：A + O = O + A = A。用阶数相容的零矩阵相乘也得到零矩阵：A × O = O（以及 O × A = O）。
The identity matrix I is a square matrix with 1s on the leading diagonal and 0s elsewhere. The 2×2 identity matrix is I = [1 0; 0 1]. It acts like the number 1: A I = I A = A (provided the products are defined).
单位矩阵 I 是一个对角线上为 1、其余位置为 0 的方阵。2×2 单位矩阵为 I = [1 0; 0 1]。它的作用类似于数字 1：A I = I A = A（只要乘积有定义）。
7. Determinant of a 2×2 Matrix | 2×2 矩阵的行列式
The determinant of a 2×2 matrix A = [a b; c d] is a scalar value given by det(A) = ad – bc. It is sometimes written as |A|.
2×2 矩阵 A = [a b; c d] 的行列式是一个标量，计算公式为 det(A) = ad – bc，有时也记作 |A|。
det[a b; c d] = ad – bc
det[a b; c d] = ad – bc
A matrix is called singular if its determinant is zero; singular matrices do not have an inverse. If det(A) ≠ 0, the matrix is non‑singular and an inverse exists.
若行列式为零，该矩阵称为 奇异矩阵；奇异矩阵没有逆矩阵。若 det(A) ≠ 0，则矩阵非奇异，逆矩阵存在。
8. Inverse of a 2×2 Matrix | 2×2 矩阵的逆矩阵
For a non‑singular 2×2 matrix A = [a b; c d], the inverse A⁻¹ is given by:
对于非奇异的 2×2 矩阵 A = [a b; c d]，其逆矩阵 A⁻¹ 由下式给出：
A⁻¹ = (1 / det(A)) × [d –b; –c a]
A⁻¹ = (1 / det(A)) × [d –b; –c a]
Always check that det(A) ≠ 0 before calculating the inverse. You can verify your answer by multiplying A by A⁻¹ to get the identity matrix I.
计算逆矩阵前务必先确认 det(A) ≠ 0。可以通过计算 A × A⁻¹ 是否得到单位矩阵 I 来验证答案。
Example: If A = [4 7; 2 6], then det(A) = 24 – 14 = 10, so A⁻¹ = (1/10) × [6 –7; –2 4] = [0.6 –0.7; –0.2 0.4].
示例：若 A = [4 7; 2 6]，则 det(A) = 24 – 14 = 10，于是 A⁻¹ = (1/10) × [6 –7; –2 4] = [0.6 –0.7; –0.2 0.4]。
9. Using Matrices to Solve Equations | 利用矩阵解方程组
A pair of simultaneous linear equations can be written in matrix form: AX = B, where A contains the coefficients, X is the column vector of variables, and B is the constant terms vector.
一对联立线性方程可以写成矩阵形式：AX = B，其中 A 包含系数，X 是变量列向量，B 是常数项列向量。
[a b; c d] [x; y] = [e; f]
[a b; c d] [x; y] = [e; f]
If A is non‑singular, multiply both sides on the left by A⁻¹ to obtain X = A⁻¹B. This gives the unique solution.
若 A 非奇异，将等式两边左乘 A⁻¹ 得到 X = A⁻¹B，即可得出唯一解。
Worked example: Solve 2x + 3y = 7, 4x + 5y = 9. The matrix equation is [2 3; 4 5] [x; y] = [7; 9]. det = 10 – 12 = –2; inverse = (–1/2) × [5 –3; –4 2] = [–2.5 1.5; 2 –1]. Then [x; y] = [–2.5×7 + 1.5×9; 2×7 + (–1)×9] = [–17.5+13.5; 14–9] = [–4; 5].
例题：解方程组 2x + 3y = 7, 4x + 5y = 9。矩阵方程为 [2 3; 4 5] [x; y] = [7; 9]。det = 10 – 12 = –2；逆矩阵 = (–1/2) × [5 –3; –4 2] = [–2.5 1.5; 2 –1]。于是 [x; y] = [–2.5×7 + 1.5×9; 2×7 + (–1)×9] = [–17.5+13.5; 14–9] = [–4; 5]。
10. Transformation Matrices – Basics | 变换矩阵基础
In IGCSE AQA, matrices are used to transform points and shapes. A point with coordinates (x, y) is written as a column vector [x; y]. Multiplying this vector by a 2×2 transformation matrix gives the new coordinates of the image.
在 IGCSE AQA 中，矩阵用于对点和图形进行变换。坐标为 (x, y) 的点写作列向量 [x; y]。将此向量左乘一个 2×2 变换矩阵，便得到像的新坐标。
Below are the standard transformation matrices for the unit square and any point:
以下是作用于单位正方形及任意点的标准变换矩阵：
| Transformation | Matrix |
|---|---|
| Reflection in the x‑axis | [1 0; 0 –1] |
| Reflection in the y‑axis | [–1
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