📚 Matrix Arithmetic for AQA A-Level | AQA A-Level 矩阵运算
Matrix arithmetic is one of the most fundamental topics in AQA A-Level Mathematics. It includes matrix addition, subtraction, scalar multiplication, matrix multiplication, transposition, determinants and finding inverses. These operations are used to represent linear transformations, solve simultaneous equations, and model real-world situations. This revision guide walks through every operation you need for the exam, with clear notation, worked examples and the common errors that cost marks.
矩阵运算是 AQA A-Level 数学中最基础的主题之一，包括矩阵加法、减法、标量乘法、矩阵乘法、转置、行列式以及求逆矩阵。这些运算用于表示线性变换、求解联立方程组以及对现实问题进行建模。本复习指南将带你梳理考试所需的每一种运算，并提供清晰的记法、例题讲解以及最容易丢分的常见错误。
1. Matrix Notation and Order | 矩阵记法与阶数
A matrix is a rectangular array of numbers arranged in rows and columns. The order (or size) of a matrix is written m × n, where m is the number of rows and n 3 matrix has 2 rows and 3 columns. The order is always quoted as rows first, then columns.
矩阵是按行和列排列的矩形数字阵列。矩阵的阶数（也称大小）写作 m × n，其中 m 表示行数，n 表示列数。例如，一个 2 × 3 矩阵有 2 行和 3 列。阶数永远先写行数，再写列数。
Matrices are usually denoted by capital letters such as A, B and M. In this guide we write [ a b ; c d ] for a 2 × 2 matrix with rows [ a b ] and [ c d ]. The semicolon separates rows, so the notation stays compact in plain text.
通常用大写字母 A、B 和 M 表示矩阵。在本指南中，我们用 [ a b ; c d ] 表示一个 2 × 2 矩阵，其两行分别为 [ a b ] 和 [ c d ]。分号用于分隔行，这样在纯文本中也能保持写法简洁。
A = [ 2 −1 ; 3 4 ] has order 2 × 2
Each number in a matrix is called an element or entry. The element in row i and column j is often written as aᵢⱼ. So in the matrix above, a₁₁ = 2, a₁₂ = −1, a₂₁ = 3 and a₂₂ = 4.
矩阵中的每个数称为元素或项。第 i 行第 j 列的元素通常写作 aᵢⱼ。因此在上面的矩阵中，a₁₁ = 2，a₁₂ = −1，a₂₁ = 3，a₂₂ = 4。
2. Addition and Subtraction of Matrices | 矩阵的加法与减法
Two matrices can be added or subtracted only if they have exactly the same order. You add or subtract the corresponding elements one by one. If the orders differ, the operation is undefined and you cannot proceed.
只有阶数完全相同的两个矩阵才能相加或相减。你需要逐个对应地相加或相减元素。如果阶数不同，该运算就是无定义的，无法继续计算。
If A = [ p q ; r s ] and B = [ u v ; w x ], then A + B = [ p+u q+v ; r+w s+x ]
如果 A 和 B 同阶，则 A + B 的每个元素就是 A 与 B 中对应位置元素之和；A − B 则是对应元素之差。减法可以理解为加上负矩阵，即 A − B = A + (−1)B。
Example: A = [ 3 5 ; 1 2 ] and B = [ 2 −1 ; 0 4 ] gives A + B = [ 5 4 ; 1 6 ] and A − B = [ 1 6 ; 1 −2 ].
例如：A = [ 3 5 ; 1 2 ]，B = [ 2 −1 ; 0 4 ]，则 A + B = [ 5 4 ; 1 6 ]，A − B = [ 1 6 ; 1 −2 ]。
Addition of matrices is commutative and associative: A + B = B + A and (A + B) + C = A + (B + C). These properties make matrix addition behave like ordinary number addition, which is why it is the easiest operation to handle.
矩阵加法满足交换律与结合律：A + B = B + A，且 (A + B) + C = A + (B + C)。这些性质使矩阵加法与普通数字加法十分相似，因此它是最容易处理的运算。
3. Scalar Multiplication | 标量乘法
A scalar is a single number (not a matrix). To multiply a matrix by a scalar, multiply every element of the matrix by that number. The order of the matrix never changes under scalar multiplication.
标量是指一个单独的数（而不是矩阵）。用标量乘矩阵时，要把矩阵中的每一个元素都乘以这个数。在标量乘法下，矩阵的阶数绝不会改变。
k[ a b ; c d ] = [ ka kb ; kc kd ]
例如，若 k = 3 且 A = [ 2 −1 ; 3 4 ]，则 3A = [ 6 −3 ; 9 12 ]，而 −A = [ −2 1 ; −3 −4 ]。
Example: with k = 3 and A = [ 2 −1 ; 3 4 ], we get 3A = [ 6 −3 ; 9 12 ], while −A = [ −2 1 ; −3 −4 ].
Scalar multiplication distributes over matrix addition: k(A + B) = kA + kB and (k + l)A = kA + lA. It also combines with matrix multiplication so that k(AB) = (kA)B = A(kB), a rule that appears regularly in AQA algebra questions.
标量乘法对矩阵加法满足分配律：k(A + B) = kA + kB，且 (k + l)A = kA + lA。它与矩阵乘法结合时还满足 k(AB) = (kA)B = A(kB)，这条规则在 AQA 的代数题中经常出现。
4. Matrix Multiplication: The Row-by-Column Rule | 矩阵乘法：行乘列法则
Two matrices can be multiplied only when the number of columns of the first matrix equals the number of rows of the second. If A has order m × n and B has order n × p, then the product AB has order m × p. The inner dimensions must match; the outer dimensions give the order of the answer.
只有当第一个矩阵的列数等于第二个矩阵的行数时，两个矩阵才能相乘。若 A 的阶数为 m × n，B 的阶数为 n × p，则乘积 AB 的阶数为 m × p。内层维度必须相同，而外层维度决定了结果的阶数。
To find the element in row i and column j of the product, you take row i of the first matrix and column j of the second, multiply the matching entries and add the results. This is the row-by-column rule.
要得到乘积中第 i 行第 j 列的元素，就取第一个矩阵的第 i 行与第二个矩阵的第 j 列，将对应元素相乘后求和。这就是"行乘列"法则。
(AB)ᵢⱼ = aᵢ₁b₁ⱼ + aᵢ₂b₂ⱼ + … + aᵢₙbₙⱼ
矩阵乘法要求对应位置元素相乘再求和，因此它比加法复杂得多，也是最容易出错的运算。
Worked example: A = [ 2 3 ; 1 4 ] and B = [ 5 1 ; 2 3 ]. Then AB = [ 2×5 + 3×2, 2×1 + 3×3 ; 1×5 + 4×2, 1×1 + 4×3 ] = [ 16 11 ; 13 13 ].
例题：A = [ 2 3 ; 1 4 ]，B = [ 5 1 ; 2 3 ]。则 AB = [ 2×5 + 3×2, 2×1 + 3×3 ; 1×5 + 4×2, 1×1 + 4×3 ] = [ 16 11 ; 13 13 ]。
Matrix multiplication is associative and distributive: (AB)C = A(BC) and A(B + C) = AB + AC. However, it is not commutative in general: AB is usually different from BA. A quick check with A = [ 1 1 ; 0 1 ] and B = [ 1 0 ; 1 1 ] gives AB = [ 2 1 ; 1 1 ] but BA = [ 1 1 ; 1 2 ].
矩阵乘法满足结合律与分配律：(AB)C = A(BC)，A(B + C) = AB + AC。但一般情况下它不满足交换律：AB 与 BA 通常不相等。用 A = [ 1 1 ; 0 1 ]、B = [ 1 0 ; 1 1 ] 快速验证可得 AB = [ 2 1 ; 1 1 ]，而 BA = [ 1 1 ; 1 2 ]。
5. The Identity Matrix and the Zero Matrix | 单位矩阵与零矩阵
The identity matrix I is the matrix analogue of the number 1. For 2 × 2 matrices it is I = [ 1 0 ; 0 here. Multiplying any square matrix by the identity of the same order leaves it unchanged: AI = IA = A.
单位矩阵 I 相当于数字中的 1。对于 2 × 2 矩阵，I = [ 1 0 ; 0 1 ]，即主对角线上是 1，其余位置是 0。任何同阶方阵乘以单位矩阵都保持不变：AI = IA = A。
The zero matrix O has every element equal to zero. It behaves like the number 0 for addition: A + O = A. Be careful, though, because AB = O does not imply that A = O or B = O, which surprises many students.
零矩阵 O 的所有元素都是 0，它在加法中相当于数字 0：A + O = A。但要小心，AB = O 并不能推出 A = O 或 B = O，这一点常常令学生感到意外。
Powers of a square matrix are defined by repeated multiplication: A² = AA, A³ = AAA, and by convention A⁰ = I where A is invertible. AQA questions often ask for A² or A³ as part of a transformation or induction-style problem.
方阵的幂由重复相乘定义：A² = AA，A³ = AAA，并约定对可逆矩阵 A 有 A⁰ = I。AQA 题目经常要求计算 A² 或 A³，作为变换题或归纳型问题的一部分。
6. The Transpose of a Matrix | 矩阵的转置
The transpose of a matrix A, written Aᵀ, is obtained by interchanging its rows and columns. If A has order m × n, then Aᵀ has order n × m. So transposing flips the shape as well as the entries.
矩阵 A 的转置记作 Aᵀ，通过交换其行与列得到。若 A 的阶数为 m × n，则 Aᵀ 的阶数为 n × m。因此转置不仅交换元素，也改变了矩阵的形状。
If A = [ a b ; c d ] then Aᵀ = [ a c ; b d ]
例如，若 A = [ 2 −1 ; 3 4 ]，则 Aᵀ = [ 2 3 ; −1 4 ]。注意主对角线上的元素 2 与 4 位置不变，而 −1 与 3 交换了位置。
Example: if A = [ 2 −1 ; 3 4 ], then Aᵀ = [ 2 3 ; −1 ]. Notice that the diagonal elements 2 and 4 stay in place while −1 and 3 swap positions.
Two useful properties are (Aᵀ)ᵀ = A and (AB)ᵀ = BᵀAᵀ. The reversal of order in the second identity is a classic exam trap. A square matrix with Aᵀ = A is called symmetric, and one with Aᵀ = −A is called skew-symmetric.
两条常用性质是 (Aᵀ)ᵀ = A 与 (AB)ᵀ = BᵀAᵀ。第二条中的次序颠倒正是经典考点。若方阵满足 Aᵀ = A，称为对称矩阵；若满足 Aᵀ = −A，称为反对称矩阵。
7. Determinant of a 2 × 2 Matrix | 2 × 2 矩阵的行列式
single number that tells you whether the matrix is invertible and how it scales areas. For A = [ a b ; c d ], the determinant is written det A or | A |.
2 × 2 矩阵的行列式是一个数，它告诉你矩阵是否可逆，以及它把面积放大或缩小了多少倍。对于 A = [ a b ; c d ]，行列式写作 det A 或 | A |。
det A = | A | = ad − bc
计算方法是主对角线元素之积减去副对角线元素之积，即 ad − bc。记忆口诀是"主对角线减副对角线"。
Worked example: for A = [ 4 7 ; 2 6 ], det A = 24 − 14 = 10. Since 10 ≠ 0, the matrix is non-singular and therefore invertible. If det A = 0, the matrix is singular and no inverse exists.
例题：A = [ 4 7 ; 2 6 ]，则 det A = 4×6 − 7×2 = 24 − 14 = 10。因为 10 ≠ 0，所以该矩阵非奇异，可逆。若 det A = 0，则矩阵奇异，不存在逆矩阵。
Useful determinant rules include det(AB) = det A × det B, det(Aᵀ) = det A, and det(kA) = k² det A for a 2 × 2 matrix. The last one catches students who forget to square the scalar.
常用行列式法则包括 det(AB) = det A × det B、det(Aᵀ) = det A，以及对 2 × 2 矩阵有 det(kA) = k² det A。最后一条常让忘记把标量平方的学生失分。
8. Inverse of a 2 × 2 Matrix | 2 × 2 矩阵的逆
The inverse of a square matrix A, written A⁻¹, satisfies AA⁻¹ = A⁻¹A = I. It behaves like a reciprocal for matrices and is the key tool for undoing a transformation or solving a matrix equation.
方阵 A 的逆矩阵记作 A⁻¹，满足 AA⁻¹ = A⁻¹A = I。它相当于矩阵中的"倒数"，是撤销变换或解矩阵方程的关键工具。
A⁻¹ = (1 ÷ det A) × [ d −b ; −c a ], valid when det A ≠ 0
求逆步骤：先交换主对角线上的 a 与 d，再把副对角线上的 b 与 c 变号，最后整体除以行列式。只有当 det A ≠ 0 时逆矩阵才存在。
Worked example: for A = [ 4 7 ; 2 6 ], det A = 10, so A⁻¹ = (1 ÷ 10)[ 6 −7 ; −2 4 ] = [ 0.6 −0.7 ; −0.2 0.4 ]. Check: AA⁻¹ = [ 4×0.6 + 7×(−0.2), 4×(−0.7) + 7×0.4 ; 2×0.6 + 6×(−0.2), 2×(−0.7) + 6×0.4 ] = [ 1 0 ; 0 1 ].
例题：A = [ 4 7 ; 2 6 ]，det A = 10，所以 A⁻¹ = (1 ÷ 10)[ 6 −7 ; −2 4 ] = [ 0.6 −0.7 ; −0.2 0.4 ]。验证：AA⁻¹ = [ 4×0.6 + 7×(−0.2), 4×(−0.7) + 7×0.4 ; 2×0.6 + 6×(−0.2), 2×(−0.7) + 6×0.4 ] = [ 1 0 ; 0 1 ]。
Other rules worth memorising are (AB)⁻¹ = B⁻¹A⁻¹, (Aᵀ)⁻¹ = (A⁻¹)ᵀ and (A⁻¹)⁻¹ = A. Again, note the reversed order in the first identity.
其他值得记住的法则有 (AB)⁻¹ = B⁻¹A⁻¹、(Aᵀ)⁻¹ = (A⁻¹)ᵀ 以及 (A⁻¹)⁻¹ = A。请注意第一条中次序同样颠倒了。
9. Matrix Arithmetic and Linear Transformations | 矩阵运算与线性变换
transformation of the plane, mapping a column vector (x ; y) to a new vector. The columns of the matrix are the images of the unit vectors (1 ; 0) and (0 ; 1), which is a fast way to construct a matrix from a description.
每个 2 × 2 矩阵都表示平面上的一个线性变换，把列向量 (x ; y) 映射为一个新向量。矩阵的两列分别是单位向量 (1 ; 0) 与 (0 ; 1) 的像，这是根据文字描述快速写出矩阵的好方法。
Common transformations to know include the identity [ 1 0 ; 0 1 ], rotation anticlockwise through θ given by [ cos θ −sin θ ; sin θ cos θ ], reflection in the y-axis [ −1 0 ; 0 1 ], and enlargement with scale factor k [ k 0 ; 0 k ].
必须掌握的常见变换包括：恒等变换 [ 1 0 ; 0 1 ]；逆时针旋转 θ 角 [ cos θ −sin θ ; sin θ cos θ ]；关于 y 轴反射 [ −1 0 ; 0 1 ]；以及比例因子为 k 的位似放大 [ k 0 ; 0 k ]。
If a transformation is represented by A and another by B, applying A first and then B is represented by the single matrix BA. This is why the order matters so much and why AB ≠ BA has a genuine geometric meaning.
若变换 A 与变换 B 分别由矩阵表示，则"先做 A 再做 B"由单个矩阵 BA 表示。这正是次序如此重要、以及 AB ≠ BA 具有真实几何意义的原因。
The area scale factor of the transformation with matrix A is | det A |. A negative determinant indicates that the orientation of the plane has been reversed, for example by a reflection.
矩阵 A 所表示变换的面积比例因子是 | det A |。行列式为负则说明平面的取向被反转，例如发生了反射。
10. Solving Simultaneous Equations with Matrices | 用矩阵求解联立方程组
A pair of simultaneous linear equations can be written as a single matrix equation. The system ax + by = e
