📚 A-Level AQA Maths: Matrices – Key Points Revision | A-Level AQA 数学：矩阵 考点精讲
Matrices are a fundamental topic in A-Level Further Mathematics (AQA). They provide a powerful way to handle linear transformations, solve systems of equations, and represent geometric operations. This revision guide covers all key concepts required for the exam, including matrix arithmetic, determinants, inverses, and transformations.
矩阵是 A-Level 进阶数学（AQA）中的一个核心主题。它们为处理线性变换、求解方程组以及表示几何运算提供了强大的方法。本复习指南涵盖了考试所需的所有关键概念，包括矩阵算术、行列式、逆矩阵和变换。
1. Definition and Order of Matrices | 矩阵的定义与阶
A matrix is a rectangular array of numbers, called elements, arranged in m rows and n matrix has 2 rows and 3 columns. The element in the ith row and jth column is denoted by aij.
矩阵是一个由数字（称为元素）组成的矩形阵列，按 m 行 n 列排列。它的阶（或维数）写作 m × n。例如，一个 2×3 矩阵有 2 行 3 列。第 i 行第 j 列的元素记作 aij。
If A is the matrix below, then a21 = 4 and a12 = -1.
若 A 是下面的矩阵，则 a21 = 4，a12 = -1。
| 3 | -1 | 0 |
| 4 | 2 | 5 |
Two matrices are equal if they have the same order and corresponding elements are equal.
两个矩阵若同阶且对应元素相等，则它们相等。
2. Matrix Addition and Subtraction | 矩阵的加法与减法
can be added or subtracted by adding or subtracting the corresponding elements. If A = (aij) and B = (bij) are both m × n, then A + B = (aij + bij) and A - B = (aij - bij).
同阶矩阵可以相加或相减，只需将对应元素相加或相减。若 A = (aij) 和 B = (bij) 均为 m × n，则 A + B = (aij + bij)，A - B = (aij - bij)。
operations are commutative and associative: A + B = B + A, and (A + B) + C = A + (B + C).
此运算满足交换律和结合律：A + B = B + A，(A + B) + C = A + (B + C)。
3. Scalar Multiplication | 标量乘法
a scalar (a constant) by multiplying every element of the matrix by that scalar. If A = (aij) and k is a scalar, then kA = (kaij).
任何矩阵都可以乘以一个标量（一个常数），只需将矩阵的每一个元素乘以该标量。若 A = (aij) 且 k 是一个标量，则 kA = (kaij)。
-3; 0 9]. Scalar multiplication is distributive over matrix addition: k(A + B) = kA + kB.
例如，若 A = [2 -1; 0 3]，则 3A = [6 -3; 0 9]。标量乘法对矩阵加法满足分配律：k(A + B) = kA + kB。
4. Matrix Multiplication | 矩阵乘法
x. If A is m × n and B is n × p, then the product C = AB is an m × p matrix. The element cij is found by multiplying elements of the ith row of A with the jth column of B and summing:
只有当第一个矩阵的列数等于第二个矩阵的行数时，矩阵乘法才有定义。若 A 是 m × n，B 是 n × p，则乘积 C = AB 是 m × p 矩阵。元素 cij 由 A 的第 i 行元素与 B 的第 j 列元素对应相乘再求和得到：
cij = ai1b1j + ai2b2j + … + ainbnj
l AB ≠ BA. However, it is associative: (AB)C = A(BC) and distributive: A(B + C) = AB + AC.
矩阵乘法不满足交换律：一般来说 AB ≠ BA。但满足结合律：(AB)C = A(BC) 和分配律：A(B + C) = AB + AC。
Worked example: multiply A = [1 2; 3 4] and B = [5 6; 7 8].
练习题：计算 A = [1 2; 3 4] 与 B = [5 6; 7 8] 的乘积。
AB = [1×5+2×7 1×6+2×8; 3×5+4×7 3×6+4×8] = [19 22; 43 50].
AB = [1×5+2×7 1×6+2×8; 3×5+4×7 3×6+4×8] = [19 22; 43 50]。
5. The Zero and Identity Matrices | 零矩阵与单位矩阵
l elements equal to 0. For any matrix A of appropriate size, A + 0 = A and A0 = 0, 0A = 0.
零矩阵记作 0，所有元素均为 0。对于任意适当大小的矩阵 A，有 A + 0 = A，且 A0 = 0，0A = 0。
atrix A, AI = IA = A. The identity matrix acts like the number 1 in matrix multiplication.
单位矩阵记作 I，是一个主对角线元素为 1、其余元素为 0 的方阵。对于 2×2 矩阵，I = [1 0; 0 1]。对于任意方阵 A，有 AI = IA = A。单位矩阵在矩阵乘法中的作用类似于数字 1。
6. Transpose of a Matrix | 矩阵的转置
The transpose of a matrix A, written AT, is obtained by interchanging rows and columns. If A is m × n, then AT is n × m with element (AT)ij = aji.
矩阵 A 的转置记作 AT，通过交换行和列得到。若 A 是 m × n，则 AT 是 n × m，且有 (AT)ij = aji。
Properties: (AT)T = A; (A + B)T = AT + BT; (kA)T = kAT; (AB)T = BTAT (note the reversal of order).
性质：(AT)T = A；(A + B)T = AT + BT；(kA)T = kAT；(AB)T = BTAT（注意顺序颠倒）。
A square matrix is symmetric if AT = A, and skew-symmetric if AT = -A.
方阵若满足 AT = A 则对称，若满足 AT = -A 则反对称。
7. Determinant and Singular Matrices | 行列式与奇异矩阵
The determinant of a 2×2 matrix A = [a b; c d] is denoted det A or |A| and calculated as ad - bc.
2×2 矩阵 A = [a b; c d] 的行列式记作 det A 或 |A|，计算公式为 ad - bc。
det A = ad - bc
lar and does not have an inverse. If det A ≠ 0, the matrix is non-singular and an inverse exists.
若 det A = 0，则该矩阵称为奇异矩阵，不存在逆矩阵。若 det A ≠ 0，则为非奇异矩阵，逆矩阵存在。
The determinant also gives the area scale factor of the transformation represented by the matrix. For example, under matrix transformation, the area is multiplied by |det A|.
行列式还给出了由矩阵表示的变换的面积缩放因子。例如，在矩阵变换下，面积乘以 |det A|。
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 the formula:
对于非奇异 2×2 矩阵 A = [a b; c d]，逆矩阵 A-1 由以下公式给出：
A-1 = (1 / det A) [d -b; -c a]
er the steps: swap a and d, change the signs of b and c, and multiply by the reciprocal of the determinant.
这里 det A = ad - bc ≠ 0。必须记住步骤：交换 a 和 d，改变 b 和 c 的符号，然后乘以行列式的倒数。
The inverse satisfies AA-1 = A-1A = I. If a matrix has an inverse, it is unique.
逆矩阵满足 AA-1 = A-1A = I。若矩阵有逆，则逆矩阵唯一。
Example: Find the inverse of [2 3; 1 4]. det = 2×4 - 3×1 = 5, so A-1 = (1/5) [4 -3; -1 2] = [0.8 -0.6; -0.2 0.4].
示例：求 [2 3; 1 4] 的逆。行列式 = 2×4 - 3×1 = 5，故 A-1 = (1/5) [4 -3; -1 2] = [0.8 -0.6; -0.2 0.4]。
9. Solving Linear Equations Using Inverse Matrices | 用逆矩阵解线性方程组
e A is the coefficient matrix, X is the column matrix of variables, and B is the constants column. Multiply both sides on the left by A-1 to obtain X = A-1B.
含有两个线性方程的方程组可写成矩阵形式 AX = B，其中 A 为系数矩阵，X 为变量列矩阵，B 为常数项列矩阵。两边左乘 A-1 得 X = A-1B。
For example, solve 2x + y = 5, x - 3y = -1. Write as [2 1; 1 -3] [x; y] = [5; -1]. A-1 = 1/(2×(-3)-1×1) [-3 -1; -1 2] = (-1/7) [-3 -1; -1 2]? Let us compute: det = -6 - 1 = -7. So A-1 = (1/-7) [-3 -1; -1 2] = [3/7 1/7; 1/7 -2/7]. Then X = A-1B gives the solution.
例如，解方程组 2x + y = 5，x - 3y = -1。写成 [2 1; 1 -3] [x; y] = [5; -1]。A-1 = 1/(-7) [-3 -1; -1 2] = [3/7 1/7; 1/7 -2/7]。然后 X = A-1B 得出解。
This method works for any number of equations where A is invertible. Always check det A ≠ 0 before finding the inverse.
该方法适用于任意方程个数，只要 A 可逆。在求逆之前务必检验 det A ≠ 0。
10. Matrix Transformations: Reflections | 矩阵变换：反射
ansformation of the plane. Reflection matrices are particularly common in the AQA syllabus. The standard reflection matrices are:
2×2 矩阵可表示平面的线性变换。反射矩阵在 AQA 考纲中尤为常见。标准反射矩阵如下：
- Reflection in the x-axis: [1 0; 0 -1] (y-coordinate changes sign)
- Reflection in the y-axis: [-1 0; 0 1] (x-coordinate changes sign)
- Reflection in the line y = x: [0 1; 1 0] (coordinates swap)
- Reflection in the line y = -x: [0 -1; -1 0]
- 关于 x 轴反射：[1 0; 0 -1]（y 坐标变号）
- 关于 y 轴反射：[-1 0; 0 1]（x 坐标变号）
- 关于直线 y = x 反射：[0 1; 1 0]（坐标互换）
- 关于直线 y = -x 反射：[0 -1; -1 0]
You should be able to apply these to points or shapes and find the image. For instance, multiplying the reflection matrix by the column vector of a point gives its image.
你应该能够将这些矩阵应用于点或图形并求出像。例如，将反射矩阵乘以点的列向量即得其像。
11. Matrix Transformations: Rotations and Enlargements | 矩阵变换：旋转与缩放
A rotation about the origin through angle θ anticlockwise is represented by the matrix:
绕原点逆时针旋转角 θ 的变换矩阵为：
R = [cos θ -sin θ; sin θ cos θ]
For example, rotation by 90° (π/2) gives [0 -1; 1 0]. Rotation by 180° gives [-1 0; 0 -1], which is also a central enlargement with scale factor -1.
例如，旋转 90°（π/2）得 [0 -1; 1 0]。旋转 180° 得 [-1 0; 0 -1]，这也是比例因子为 -1 的中心放大。
An enlargement (scaling) with centre at the origin and scale factor k is represented by [k 0; 0 k]. This is a diagonal matrix, and it preserves shape but changes area by a factor of k².
以原点为中心、比例因子为 k 的缩放变换由 [k 0; 0 k] 表示。这是一个对角矩阵，它保持形状但将面积缩放 k² 倍。
Stretches parallel to the axes can be represented by matrices like [a 0; 0 1] or [1 0; 0 b]. These are useful for shear or one-way stretching.
平行于坐标轴的拉伸可由 [a 0; 0 1] 或 [1 0; 0 b] 等矩阵表示。这些对于剪切或单向拉伸有用。
12. Composite Transformations and Matrix Equations | 复合变换与矩阵方程
When multiple transformations are applied in succession, the overall matrix is the product of the individual matrices, applied from right to left. If transformation A is followed by transformation B, the combined matrix is BA (B first applied on the left of A's vector).
当相继应用多个变换时，整体矩阵是各个矩阵的乘积，且按从右到左的顺序作用。若先进行变换 A 再进行变换 B，则复合矩阵为 BA（B 左乘 A 作用后的向量）。
Example: reflect in the y-axis then rotate
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