Matrices | 矩阵

📚 Matrices | 矩阵

Matrices are a fundamental topic in A-Level mathematics. They provide a compact way to solve simultaneous equations and to describe linear transformations of the plane. This revision guide covers the essential matrix skills required for the AQA A-Level Mathematics course.

矩阵是 A-Level 数学中的一个基础主题。它提供了解联立方程的简洁方法，并用于描述平面上的线性变换。本复习指南涵盖 AQA A-Level 数学课程所需的矩阵核心技能。

1. What is a Matrix? | 什么是矩阵？

A matrix is a rectangular array of numbers arranged in rows and columns. The order of a matrix is written as “rows × columns”. For example, a 2 × 3 matrix has 2 rows and 3 columns:

矩阵是按行和列排列的矩形数字数组。矩阵的阶写作“行数 × 列数”。例如，一个 2 × 3 矩阵有 2 行 3 列：

A = [1 2 3; 4 5 6] (order 2 × 3)

Entries are referred to by their row and column positions; aᵢⱼ denotes the entry in row i and column j. In this example, a₁₂ = 2 and a₂₃ = 6.

矩阵中的元素按其行和列的位置引用；aᵢⱼ 表示第 i 行第 j 列的元素。在此例中，a₁₂ = 2，a₂₃ = 6。

A matrix with an equal number of rows and columns is called a square matrix. The main diagonal runs from top-left to bottom-right. Many AQA questions focus on 2 × 2 square matrices.

行数和列数相等的矩阵称为方阵。主对角线从左上延伸到右下。AQA 考试题往往聚焦于 2 × 2 方阵。

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2. Addition, Subtraction and Scalar Multiplication | 加法、减法与标量乘法

Two matrices can be added or subtracted only if they have the same order. Corresponding entries are added or subtracted:

两个矩阵只有在阶数相同时才能相加或相减。对应位置的元素相加或相减：

[a b; c d] + [e f; g h] = [a+e b+f; c+g d+h]

Scalar multiplication means multiplying every entry of the matrix by the scalar. If k is a number, then kA multiplies each entry of A by k.

标量乘法是指将矩阵中的每个元素都乘以该标量。若 k 是一个数，则 kA 将 A 中的每个元素都乘以 k。

• Example: 2 × [1 2; 3 4] = [2 4; 6 8]
• 例：2 × [1 2; 3 4] = [2 4; 6 8]

These operations follow the usual rules of arithmetic: they are commutative and associative for addition.

这些运算遵循通常的算术法则：加法满足交换律和结合律。

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3. Matrix Multiplication | 矩阵乘法

Matrix multiplication is more involved. If A is m × n and B is n × p, then the product AB is defined and has order m × p. The entry in row i and column j of AB is found by multiplying each entry of row i of A by the corresponding entry of column j of B, then adding the results.

矩阵乘法更复杂。若 A 是 m × n 阶，B 是 n × p 阶，则乘积 AB 有定义，且阶数为 m × p。AB 的第 i 行第 j 列元素由 A 的第 i 行各元素与 B 的第 j 列对应元素相乘后求和得到。

(AB)ᵢⱼ = Σₖ aᵢₖ bₖⱼ

For 2 × 2 matrices,

对于 2 × 2 矩阵，

[a b; c d] × [e f; g h] = [ae+bg af+bh; ce+dg cf+dh]

Matrix multiplication is not commutative: in general AB ≠ BA. You must always be careful about the order of multiplication.

矩阵乘法不满足交换律：一般 AB ≠ BA。在计算时务必注意乘法的顺序。

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4. Identity Matrix and Determinant | 单位矩阵与行列式

The identity matrix, written I, is a square matrix with 1s on the main diagonal and 0s elsewhere. For any matrix A of the same order, AI = IA = A.

单位矩阵记作 I，是主对角线为 1、其余位置为 0 的方阵。对于任何同阶矩阵 A，AI = IA = A。

I = [1 0; 0 1]

For a 2 × 2 matrix A = [a b; c d], the determinant is defined as

对于 2 × 2 矩阵 A = [a b; c d]（即 a、b 在首行，c、d 在次行），行列式定义为

det(A) = ad − bc

The determinant is often written using straight vertical bars: |A|. It determines whether a matrix is invertible and also appears in area scale factors.

行列式常用竖直两条线表示：|A|。它决定矩阵是否可逆，也出现在面积缩放因子中。

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5. Inverse of a 2 × 2 Matrix | 2 × 2 矩阵的逆

If det(A) ≠ 0, the matrix A is non-singular and has an inverse A⁻¹ such that AA⁻¹ = A⁻¹A = I.

若 det(A) ≠ 0，则矩阵 A 非奇异，存在逆矩阵 A⁻¹ 满足 AA⁻¹ = A⁻¹A = I。

A⁻¹ = 1/(ad − bc) × [d −b; −c a]

To form the inverse of a 2 × 2 matrix: swap a and d, change the signs of b and c, then divide by the determinant. If det(A) = 0, A is singular and has no inverse.

求 2 × 2 矩阵的逆：交换 a 和 d，改变 b 和 c 的正负号，再除以行列式。若 det(A) = 0，则 A 奇异，无逆矩阵。

• Example: A = [4 3; 3 2], det(A) = 4×2 − 3×3 = −1, so A⁻¹ = [−2 3; 3 −4].
• 例：A = [4 3; 3 2]，det(A) = 4×2 − 3×3 = −1，因此 A⁻¹ = [−2 3; 3 −4]。

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6. Solving Simultaneous Equations Using Matrices | 用矩阵解联立方程

A system of two linear equations,

一个二元线性方程组，

ax + by = e, cx + dy = f

can be written in matrix form as

可以写成矩阵形式

[a b; c d] [x; y] = [e; f]

Let A = [a b; c d], X = [x; y], and B = [e; f]. If A⁻¹ exists, we multiply both sides on the left by A⁻¹ to get X = A⁻¹B.

设 A = [a b; c d]，X = [x; y]，B = [e; f]。若 A⁻¹ 存在，则将等式两边左乘 A⁻¹，得到 X = A⁻¹B。

Worked example: Solve 2x + 3y = 7, 3x + 4y = 10.

例题：解方程组 2x + 3y = 7，3x + 4y = 10。

A = [2 3; 3 4], det(A) = 2×4 − 3×3 = −1

A⁻¹ = [−4 3; 3 −2]

[x; y] = [−4 3; 3 −2] [7; 10] = [−4×7+3×10; 3×7−2×10] = [2; 1]

Therefore x = 2, y = 1. Always check your answers by substituting back into the original equations.

因此 x = 2，y = 1。务必将答案代回原方程检验。

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7. Linear Transformations | 线性变换

A 2 × 2 matrix can be used to transform a position vector [x; y] into a new vector [x′; y′] by matrix multiplication:

2 × 2 矩阵可用于将位置向量 [x; y] 变换为新向量 [x′; y′]：

[x′; y′] = [a b; c d] [x; y]

This maps points of the plane to other points, always sending the origin to itself and preserving straight lines. Such maps are called linear transformations.

这会把平面上的点映射到其他点，始终保持原点不动并保持直线为直线。这样的映射称为线性变换。

For example, the matrix [0 −1; 1 0] rotates a point anticlockwise by 90° about the origin. Test: (1, 0) goes to (0, 1).

例如，矩阵 [0 −1; 1 0] 将点绕原点逆时针旋转 90°。检验：(1, 0) 变为 (0, 1)。

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8. Combining Transformations | 复合变换

If a transformation with matrix A is followed by a transformation with matrix B, the overall transformation matrix is BA, not AB. This is because the vector is first multiplied by A, then by B:

若先进行矩阵 A 对应的变换，再进行矩阵 B 对应的变换，总体变换矩阵是 BA，而不是 AB。因为向量先乘以 A，再乘以 B：

B(Av) = (BA)v

The order of transformations is therefore crucial. When describing the resultant transformation, we write the matrices in the reverse order of application.

因此变换顺序至关重要。在描述合成变换时，我们将矩阵按应用的逆序写出。

Example: a reflection in the y-axis followed by a 90° anticlockwise rotation combines as AB where A is the rotation matrix and B is the reflection matrix.

例：先作关于 y 轴的反射，再作绕原点逆时针 90° 旋转，可表示为 AB，其中 A 是旋转矩阵，B 是反射矩阵。

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9. Determinant and Area Scale Factor | 行列式与面积缩放因子

For a linear transformation represented by a 2 × 2 matrix A, the absolute value of det(A) gives the area scale factor of the transformation. If det(A) < 0, the orientation of the shape is reversed.

对于 2 × 2 矩阵 A 表示的线性变换，|det(A)| 给出该变换的面积缩放因子。若 det(A) < 0，则图形方向被翻转。

For example, the matrix [2 0; 0 2] has determinant 4. It doubles lengths in both directions, so any shape’s area is multiplied by 4.

例如，矩阵 [2 0; 0 2] 的行列式为 4。它把两个方向上的长度都加倍，因此任意图形的面积变为原来的 4 倍。

This fact is often tested in AQA questions involving the image of a shape after a transformation.

AQA 考试中常要求利用此性质求变换后图形的面积。

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10. Special Transformation Matrices | 特殊变换矩阵

The following are common 2 × 2 transformation matrices you should know for AQA:

以下是你必须掌握的常见 2 × 2 变换矩阵：

Transformation 变换	Matrix 矩阵
Reflection in x-axis | 关于 x 轴对称	[1 0; 0 −1]
Reflection in y-axis | 关于 y 轴对称	[−1 0; 0 1]
Reflection in y = x | 关于 y = x 对称	[0 1; 1 0]
Rotation anticlockwise by θ | 逆时针旋转 θ	[cos θ −sin θ; sin θ cos θ]
Enlargement scale factor k | 放大比例 k	[k 0; 0 k]
Horizontal shear | 水平剪切	[1 k; 0 1]

You should memorise these and be able to identify the transformation from a given matrix.

你需要记住这些矩阵，并能够从给定矩阵识别对应的变换。

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11. Inverse Transformations | 逆变换

If matrix A represents a certain transformation, then A⁻¹ represents the inverse transformation that undoes it. For example, if A is a reflection in the line y = x, then A⁻¹ = A because reflecting twice returns the original point.

若矩阵 A 表示某个变换，则 A⁻¹ 表示撤销该变换的逆变换。例如，若 A 是关于直线 y = x 的反射，因为反射两次使点回到原处，所以 A⁻¹ = A。

For a rotation matrix R, the inverse is the same rotation by the opposite angle, i.e. R⁻¹ with −θ instead of θ. Since cos(−θ) = cos θ and sin(−θ) = −sin θ, the inverse matrix is the transpose of R.

对于旋转矩阵 R，其逆矩阵是旋转相反角度 −θ 的矩阵。由于 cos(−θ) = cos θ，sin(−θ) = −sin θ，因此逆矩阵是 R 的转置。

To know whether a transformation has an inverse, check that det(A) ≠ 0.

判断变换是否可逆，只需检查 det(A) ≠ 0。

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12. Exam Style Worked Example | 考试风格例题

Let’s work through a typical AQA question in full.

我们完整解答一道典型的 AQA 题目。

Question: The matrix M = [3 k; 2 1] has determinant −1. (a) Find the value of k. (b) Find M⁻¹. (c) Use M⁻¹ to solve the simultaneous equations 3x + ky = 4 and 2x + y = 7.

题目：已知矩阵 M = [3 k; 2 1] 的行列式为 −1。(a) 求 k 的值。(b) 求 M⁻¹。(c) 利用 M⁻¹ 解方程组 3x + ky = 4 和 2x + y = 7。

(a) det(M) = 3×1 − k×2 = 3 − 2k = −1, so k = 2.

(a) det(M) = 3×1 − k×2 = 3 − 2k = −1，所以 k = 2。

(b) M = [3 2; 2 1], so M⁻¹ = 1/(−1) × [1 −2; −2 3] = [−1 2; 2 −3].

(b) M = [3 2; 2 1]，所以 M⁻¹ = 1/(−1) × [1 −2; −2 3] = [−1 2; 2 −3]。

(c) Write the equations as M [x; y] = [4; 7]. Then [x; y] = M⁻¹ [4; 7] = [−1×4+2×7; 2×4−3×7] = [10; −13].

(c) 将方程组写成 M [x; y] = [4; 7]。则 [x; y] = M⁻¹ [4; 7] = [−1×4+2×7; 2×4−3×7] = [10; −13]。

Check: 2×10 + (−13) = 20 − 13 = 7, and 3×10 + 2×(−13) = 30 − 26 = 4. Hence x = 10, y = −13.

检验：2×10 + (−13) = 20 − 13 = 7；3×10 + 2×(−13) = 30 − 26 = 4。因此 x = 10，y = −13。

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