Matrices | 矩阵

📚 Matrices | 矩阵

Matrices are rectangular arrays of numbers with powerful applications in algebra, transformations, and solving equations. This revision guide covers the essential matrix operations and methods required for AQA A-Level Mathematics.

矩阵是按矩形排列的数字，具有代数、变换和解方程等强大应用。本复习指南涵盖 AQA A-Level 数学所要求的矩阵基本运算与方法。

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1. What is a Matrix? | 什么是矩阵？

A matrix (plural: matrices) is a rectangular arrangement of numbers, called entries or elements, displayed in rows and columns. For example, the matrix M = (2 3; 5 7) has two rows and two columns.

矩阵（复数：matrices）是按行和列排列的数字的矩形阵列，这些数字称为“元”或“元素”。例如，矩阵 M = (2 3; 5 7) 有 2 行和 2 列。

Each element is identified by its position. If M has entry mᵢⱼ, the subscript i indicates the row and j indicates the column. In the example above, m₁₂ = 3 because 3 is in row 1, column 2.

每个元素通过位置来识别。若 M 的元素为 mᵢⱼ，下标 i 表示行，j 表示列。在上例中，m₁₂ = 3，因为 3 位于第 1 行第 2 列。

• A matrix with m rows and n columns is called an m × n matrix (read 'm by n').

  有 m 行、n 列的矩阵称为 m × n 矩阵（读作“m 乘 n”）。

• If m = n, the matrix is called a square matrix.

  若 m = n，则矩阵称为方阵。

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2. Matrix Notation and Order | 矩阵的记号与阶数

gives the number of rows and columns, for example a 2 × 3 matrix has 2 rows and 3 columns.

矩阵通常用大写字母表示，数组外加圆括号或方括号。矩阵的“阶”表示其行数和列数，例如 2 × 3 矩阵有 2 行和 3 列。

Here are two examples:

下面给出两个例子：

A = (1 2 3; 4 5 6) is a 2 × 3 matrix.
B = (a b; c d) is a 2 × 2 matrix.

A = (1 2 3; 4 5 6) 是 2 × 3 矩阵。
B = (a b; c d) 是 2 × 2 矩阵。

The notation A = [aᵢⱼ] means that the entry in row i and column j is aᵢⱼ. For a 3 × 3 matrix, the entries are a₁₁, a₁₂, a₁₃, a₂₁, …, a₃₃.

记号 A = [aᵢⱼ] 表示第 i 行、第 j 列的元素为 aᵢⱼ。对于 3 × 3 矩阵，元素依次为 a₁₁, a₁₂, a₁₃, a₂₁, …, a₃₃。

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3. Addition and Subtraction | 矩阵的加法与减法

Two matrices can be added or subtracted only if they have the same order. The result is obtained by adding or subtracting corresponding entries.

矩阵相加或相减的前提是两个矩阵的阶相同。结果通过对相应位置元素进行相加或相减得到。

(a b; c d) + (e f; g h) = (a+e b+f; c+g d+h)

For example:

例如：

(1 2; 3 4) + (5 6; 7 8) = (6 8; 10 12)

Subtraction works in the same way: subtract each entry of the second matrix from the corresponding entry of the first.

减法同理：用第一个矩阵的每个元素减去第二个矩阵的对应元素。

• Addition is commutative: A + B = B + A.

  加法满足交换律：A + B = B + A。

• Addition is associative: (A + B) + C = A + (B + C).

  加法满足结合律：(A + B) + C = A + (B + C)。

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4. Scalar Multiplication | 数与矩阵相乘

To multiply a matrix by a scalar (a single number), multiply every entry inside the matrix by that scalar.

用数乘矩阵时，矩阵内的每一个元素都乘以这个数。

k × (a b; c d) = (ka kb; kc kd)

For example:

例如：

3 × (1 2; 0 -1) = (3 6; 0 -3)

Scalar multiplication is distributive over matrix addition: k(A + B) = kA + kB.

数乘对矩阵加法满足分配律：k(A + B) = kA + kB。

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5. Matrix Multiplication | 矩阵乘法

n the second matrix. If A is m × n and B is n × p, then AB is defined and has order m × p.

矩阵乘法只有在第一个矩阵的列数等于第二个矩阵的行数时才能进行。若 A 是 m × n，B 是 n × p，则 AB 有定义，并且结果为 m × p 阶矩阵。

The entry in row i and column j of AB is found by taking the dot product of row i of A with column j of B.

AB 的第 i 行、第 j 列的元素通过 A 的第 i 行与 B 的第 j 列对应元素相乘后求和得到。

(AB)ᵢⱼ = Σₖ aᵢₖ bₖⱼ

For two 2 × 2 matrices:

对于两个 2 × 2 矩阵：

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

Note that matrix multiplication is not commutative: in general AB ≠ BA.

注意矩阵乘法不满足交换律：一般地 AB ≠ BA。

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6. Determinants | 行列式

The determinant is a scalar value associated with a square matrix. It tells us whether a matrix is invertible and also appears in area/volume transformations.

行列式是与方阵关联的一个数值。它可以判断矩阵是否可逆，也出现在面积/体积变换中。

For a 2 × 2 matrix A = (a b; c d), the determinant is:

对于 2 × 2 矩阵 A = (a b; c d)，其行列式为：

det(A) = |A| = ad − bc

trix is singular (has no inverse). If det(A) ≠ 0, the matrix is non-singular (invertible).

若 det(A) = 0，矩阵是奇异的（没有逆矩阵）。若 det(A) ≠ 0，矩阵是非奇异的（可逆）。

3 matrix, the determinant can be found by expansion along a row or column. For example, if

对于 3 × 3 矩阵，可以按某一行或某一列展开求行列式。例如，设

M = (a b c; d e f; g h i)

then det(M) = a(ei − fh) − b(di − fg) + c(dh − eg).

则 det(M) = a(ei − fh) − b(di − fg) + c(dh − eg)。

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7. Inverse of a Matrix | 逆矩阵

matrix A is denoted A⁻¹, and satisfies A A⁻¹ = A⁻¹ A = I, where I is the identity matrix.

方阵 A 的逆矩阵记作 A⁻¹，满足 A A⁻¹ = A⁻¹ A = I，其中 I 是单位矩阵。

For a 2 × 2 matrix A = (a b; c d), the inverse is:

对于 2 × 2 矩阵 A = (a b; c d)，其逆矩阵为：

A⁻¹ = 1/(ad−bc) × (d −b; −c a)

This exists only if ad − bc ≠ 0.

该逆矩阵仅在 ad − bc ≠ 0 时存在。

In AQA, you are often expected to know the 2 × 2 case and the condition for invertibility.

对于 3 × 3 矩阵，逆矩阵的计算更为复杂，通常使用伴随矩阵法或行变换法。在 AQA 考试中，学生通常需要掌握 2 × 2 情况以及可逆条件。

• If det(A) = 0, the matrix has no inverse.

  若 det(A) = 0，矩阵没有逆矩阵。

• If det(A) ≠ 0, we say A is invertible or non-singular.

  若 det(A) ≠ 0，则称 A 可逆或非奇异。

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8. Solving Systems of Linear Equations | 用矩阵求解线性方程组

s the coefficient matrix, X is the column matrix of variables, and B is the constant column matrix.

线性方程组可以写成矩阵形式 AX = B，其中 A 是系数矩阵，X 是变量列矩阵，B 是常数项列矩阵。

If A is invertible, the solution is X = A⁻¹B.

若 A 可逆，则解为 X = A⁻¹B。

Example: solve the system

例如：解方程组

2x + 3y = 8
x − y = −1

Write as:

写为：

(2 3; 1 −1) × (x; y) = (8; −1)

Here A = (2 3; 1 −1), det(A) = 2×(−1) − 3×1 = −5 ≠ 0, so A⁻¹ exists.

这里 A = (2 3; 1 −1)，det(A) = 2×(−1) − 3×1 = −5 ≠ 0，所以 A⁻¹ 存在。

A⁻¹ = 1/(−5) × (−1 −3; −1 2) = (0.2 0.6; 0.2 −0.4). Then X = A⁻¹B gives x = 1, y = 2.

A⁻¹ = 1/(−5) × (−1 −3; −1 2) = (0.2 0.6; 0.2 −0.4)。然后 X = A⁻¹B 得到 x = 1，y = 2。

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9. Matrix Transformations | 矩阵变换

ansformation of the plane. The image of a point (x, y) under the transformation is obtained by multiplying the column vector (x; y) by the matrix.

2 × 2 矩阵可以表示平面上的线性变换。点 (x, y) 在该变换下的像，就是用列向量 (x; y) 乘以该矩阵得到的。

Common transformations:

常见的变换：

• Reflection in the x-axis: (1 0; 0 −1)

  关于 x 轴的反射：(1 0; 0 −1)

• Reflection in the y-axis: (−1 0; 0 1)

  关于 y 轴的反射：(−1 0; 0 1)

• Rotation by 90° anticlockwise: (0 −1; 1 0)

  逆时针旋转 90°：(0 −1; 1 0)

• Enlargement by scale factor k: (k 0; 0 k)

  放大 k 倍：(k 0; 0 k)

• Shear parallel to the x-axis: (1 k; 0 1)

  沿 x 轴的切变：(1 k; 0 1)

The determinant of a transformation matrix gives the scale factor of the area after the transformation. A negative determinant indicates a reflection is involved.

变换矩阵的行列式给出变换后面积的变化倍数。若行列式为负，则表示变换包含反射。

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10. Key AQA Exam Tips | AQA 考试要点

Here are the most common pitfalls and tips when working with matrices in AQA A-Level exams.

下面是在 AQA A-Level 考试中关于矩阵最常见的错误与建议。

• Always check the order of multiplication. AB and BA are usually different.

  务必检查乘法顺序。AB 与 BA 通常不同。

• Only add or subtract matrices with the same order.

  只能对同阶矩阵进行加法或减法。

• inverse formula for a 2 × 2 matrix requires det(A) ≠ 0. If the determinant is zero, write 'singular' and stop.

  2 × 2 矩阵的求逆公式要求 det(A) ≠ 0。若行列式为 0，应写出“奇异”并停止运算。

• n solving AX = B, remember to multiply both sides by A⁻¹ on the left: X = A⁻¹B, not B A⁻¹.

  解 AX = B 时，记住要在等式两边左乘 A⁻¹：X = A⁻¹B，而不是 B A⁻¹。

• Verify your inverse by checking A A⁻¹ = I.

  通过验证 A A⁻¹ = I 来检查逆矩阵是否正确。

• Learn the standard transformation matrices by heart, and know how the determinant links to area scale.

  熟记标准变换矩阵，并理解行列式与面积缩放的关系。

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