📚 GCSE AQA Maths: Matrices - Key Points | GCSE AQA 数学：矩阵考点精讲
Matrices are a compact and powerful way to handle multiple numbers and transform them efficiently. In the AQA GCSE Further Mathematics (8365) specification, you are expected to understand matrix operations up to 2 by 2, including multiplication, inverses, and using matrices to solve simultaneous equations. Mastering these concepts will not only help you tackle exam questions with confidence but also build a solid foundation for A‑level Mathematics and beyond.
矩阵是一种高效处理多个数字并进行变换的紧凑工具。AQA GCSE 进阶数学（8365）要求掌握 2×2 矩阵的运算，包括乘法、逆矩阵以及利用矩阵解联立方程组。吃透这些概念不仅能让你从容应对考题，也为 A‑level 数学打下坚实基础。
1. Introduction to Matrices | 矩阵简介
s. Matrices are usually denoted by capital letters such as A, B, M. Each number inside is called an element. Two matrices are equal only if they have the same dimensions and all corresponding elements are equal.
矩阵是由数字构成的矩形阵列，按行和列排列。矩阵的大小由行数与列数决定，例如 2×2 矩阵有 2 行和 2 列。矩阵通常用大写字母表示，如 A、B、M。里面的每个数字叫作元素。仅当两个矩阵维度相同且所有对应元素相等时，它们才相等。
For a 2 × 2 matrix we often write it as:
一个 2×2 矩阵常记作：
| a | b |
| c | d |
Element a is in row 1, column 1; b is row 1, column 2; c is row 2, column 1; d is row 2, column 2.
元素 a 在第 1 行第 1 列；b 在第 1 行第 2 列；c 在第 2 行第 1 列；d 在第 2 行第 2 列。
2. Adding and Subtracting Matrices | 矩阵加减法
You can only add or subtract matrices that have exactly the same dimensions. To find the sum A + B, simply add the elements in corresponding positions. Subtraction works in the same way: subtract corresponding elements.
只有维度完全相同的矩阵才能相加或相减。求和 A＋B 只需将对应位置的元素相加。减法同理：对应元素相减。
Example: If A =
| 2 | 5 |
| 1 | 3 |
| 4 | 0 |
| -2 | 1 |
例如：若 A =
| 2 | 5 |
| 1 | 3 |
| 4 | 0 |
| -2 | 1 |
A + B =
= 2+4 5+0 1+(-2) 3+1 6 5 -1 4
Notice that matrix addition is commutative: A + B = B + A. Subtraction is not commutative.
注意矩阵加法满足交换律：A + B = B + A。减法不满足交换律。
3. Scalar Multiplication | 标量乘法
When you multiply a matrix by a number (a scalar), every element is multiplied by that number. For a scalar k and matrix M, kM multiplies each entry by k.
用数（标量）乘矩阵时，矩阵中的每个元素都乘以该数。对于标量 k 和矩阵 M，kM 将每个元素乘以 k。
Example: 3 ×
| 1 | 4 |
| -2 | 0 |
| 3×1 | 3×4 |
| 3×(-2) | 3×0 |
| 3 | 12 |
| -6 | 0 |
Scalar multiplication is fundamental for combining with addition and for understanding linear combinations of matrices.
标量乘法是组合加法以及理解矩阵线性组合的基础。
4. Matrix Multiplication | 矩阵乘法
matrix. Each entry is found by taking the dot product of a row from A and a column from B.
矩阵乘法并不是对应元素相乘。只有当 A 的列数等于 B 的行数时，乘积 AB 才有定义。两个 2×2 矩阵相乘的结果仍然是 2×2 矩阵。每个元素由 A 的行与 B 的列的点积计算得到。
If A =
| a₁₁ | a₁₂ |
| a₂₁ | a₂₂ |
| b₁₁ | b₁₂ |
| b₂₁ | b₂₂ |
| a₁₁b₁₁ + a₁₂b₂₁ | a₁₁b₁₂ + a₁₂b₂₂ |
| a₂₁b₁₁ + a₂₂b₂₁ | a₂₁b₁₂ + a₂₂b₂₂ |
若 A =
| a₁₁ | a₁₂ |
| a₂₁ | a₂₂ |
| b₁₁ | b₁₂ |
| b₂₁ | b₂₂ |
| a₁₁b₁₁ + a₁₂b₂₁ | a₁₁b₁₂ + a₁₂b₂₂ |
| a₂₁b₁₁ + a₂₂b₂₁ | a₂₁b₁₂ + a₂₂b₂₂ |
Important: Matrix multiplication is not commutative. In general, AB ≠ BA. Always check the order carefully.
重要提示：矩阵乘法不满足交换律！通常 AB ≠ BA。务必仔细核对顺序。
5. Zero and Identity Matrices | 零矩阵与单位矩阵
the zero matrix gives a zero matrix: A × O = O and O × A = O (provided dimensions match).
零矩阵（记作 O）的所有元素都是 0。加上零矩阵不会改变原矩阵：A + O = A。零矩阵参与乘法（维度允许时）会得到零矩阵：A×O = O，O×A = O。
matrix I is a square matrix with 1s on the main diagonal and 0s elsewhere. For 2 × 2: I =
| 1 | 0 |
| 0 | 1 |
单位矩阵 I 是主对角线元素为 1、其余元素为 0 的方阵。2×2 单位矩阵为：I =
| 1 | 0 |
| 0 | 1 |
6. Transpose of a Matrix | 转置矩阵
The transpose of a matrix A, written Aᵀ is obtained by swapping rows and columns. The first row of A becomes the first column of Aᵀ, and the second row becomes the second column. For a 2 × 2 matrix:
矩阵 A 的转置记作 Aᵀ，通过交换行与列得到。A 的第一行变成 Aᵀ 的第一列，第二行变成第二列。对于 2×2 矩阵：
If A =
then Aᵀ = a b c d a c b d
Transposing twice returns the original matrix: (Aᵀ)ᵀ = A. The transpose of a product reverses the order: (AB)ᵀ = Bᵀ Aᵀ.
转置两次恢复为原矩阵：(Aᵀ)ᵀ = A。乘积的转置会颠倒顺序：(AB)ᵀ = Bᵀ Aᵀ。
7. Determinant and Inverse of a 2 × 2 Matrix | 2×2 矩阵的行列式与逆矩阵
For a 2 × 2 matrix A =
| a | b |
| c | d |
对于 2×2 矩阵 A =
| a | b |
| c | d |
When det(A) ≠ 0, the inverse matrix A⁻¹ is:
当 det(A) ≠ 0 时，逆矩阵 A⁻¹ 为：
A⁻¹ =
= (1/(ad − bc)) d / det(A) −b / det(A) −c / det(A) a / det(A) d −b −c a
Always check your answer by multiplying A A⁻¹ – you should get the identity matrix I.
最后一定要用 A A⁻¹ 验算，应得到单位矩阵 I。
8. Solving Simultaneous Equations Using the Inverse Matrix
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