GCSE CIE Maths: Matrices Explained | GCSE CIE 数学：矩阵考点精讲

📚 GCSE CIE Maths: Matrices Explained | GCSE CIE 数学：矩阵考点精讲

Matrices are rectangular arrays of numbers, symbols, or expressions arranged in rows and columns. In GCSE CIE Mathematics (Extended), matrices form a vital topic, giving you powerful methods for handling data, solving simultaneous equations, and performing geometric transformations. This revision guide covers every key concept you need to master for your exam, with clear explanations in both English and Chinese.

矩阵是由数字、符号或表达式按行和列排列而成的矩形阵列。在 GCSE CIE 数学扩展课程中，矩阵是一个极其重要的主题，它为你提供了处理数据、求解联立方程组以及进行几何变换的有力工具。这份考点精讲涵盖了你需要掌握的所有关键概念，并配有清晰的中英双语解释。

1. What are Matrices? | 什么是矩阵？

A matrix is a rectangular arrangement of numbers, called elements or entries, enclosed in brackets. Each matrix is usually named with a capital letter, such as A, B, or M.

矩阵是一个由数字（称为元素或项）构成的矩形排列，并用括号括起来。每个矩阵通常用大写字母命名，例如 A、B 或 M。

The position of an element is identified by its row and column number. The element in the i-th row and j-th column is written as a_ij. Matrices can be used to store information, represent systems of equations, or describe geometric transformations.

矩阵中元素的位置由其行数和列数确定。位于第 i 行第 j 列的元素记作 a_ij。矩阵可用于存储信息、表示方程组或描述几何变换。

For example, consider a matrix that records the numbers of footballs and basketballs sold in two shops. The arrangement makes the data easy to read and manipulate.

例如，考虑一个记录两家商店足球和篮球销售数量的矩阵。这种排列使数据易于阅读和操作。

2. Order of a Matrix | 矩阵的阶

The order of a matrix gives its size, always stated as rows × columns. A matrix with m rows and n columns has order m × n (read ‘m by n’). It is crucial to check orders before performing addition, subtraction, or multiplication.

矩阵的阶描述其尺寸，始终以 行 × 列 的形式表示。一个具有 m 行 n 列的矩阵，其阶为 m × n（读作 “m 乘 n”）。在进行加减或乘法运算前，必须先检查矩阵的阶是否匹配。

For instance, matrix A = [5 −2; 1 0; 4 3] has 3 rows and 2 columns, so its order is 3 × 2. A square matrix has equal numbers of rows and columns, such as 2 × 2 or 3 × 3.

例如，矩阵 A = [5 −2; 1 0; 4 3] 有 3 行和 2 列，因此它的阶是 3 × 2。方阵的行数和列数相等，例如 2 × 2 或 3 × 3 矩阵。

Always write the row number first, then the column number. ‘3 × 2’ is not the same as ‘2 × 3’.

总是先写行数，再写列数。”3 × 2″ 与 “2 × 3” 是不同的。
The order tells you how many elements the matrix contains (m × n elements).

矩阵的阶告诉你它包含多少个元素 (共有 m × n 个元素)。

3. Adding and Subtracting Matrices | 矩阵的加减运算

Matrices can be added or subtracted only if they have the same order. You simply add or subtract the corresponding elements in each position.

只有当矩阵具有 相同的阶 时，它们才能相加或相减。只需将每个相同位置的元素对应相加或相减即可。

If A = [a b; c d] and B = [p q; r s], then A + B = [a+p b+q; c+r d+s] and A − B = [a−p b−q; c−r d−s]. The resulting matrix has the same order as A and B.

如果 A = [a b; c d] 且 B = [p q; r s]，那么 A + B = [a+p b+q; c+r d+s]，A − B = [a−p b−q; c−r d−s]。结果矩阵的阶与 A 和 B 相同。

Matrix addition is commutative (A + B = B + A) and associative ((A + B) + C = A + (B + C)). Subtraction is neither commutative nor associative, so be careful with the signs.

矩阵加法满足交换律 (A + B = B + A) 和结合律 ((A + B) + C = A + (B + C))。减法既不满足交换律也不满足结合律，因此要特别注意符号。

Example: [2 1; 0 −3] + [−1 4; 2 5] = [2−1 1+4; 0+2 −3+5] = [1 5; 2 2].

示例：[2 1; 0 −3] + [−1 4; 2 5] = [2−1 1+4; 0+2 −3+5] = [1 5; 2 2]。

4. Scalar Multiplication | 标量乘法

A scalar is a single number. To multiply a matrix by a scalar, multiply every element of the matrix by that scalar. The order of the matrix remains unchanged.

标量就是一个单独的数。要用标量乘以矩阵，只需将矩阵中的每一个元素都乘以这个标量。矩阵的阶保持不变。

If k is a scalar and A = [a b; c d], then kA = [ka kb; kc kd]. This operation is distributive over matrix addition: k(A + B) = kA + kB.

如果 k 是一个标量，且 A = [a b; c d]，那么 kA = [ka kb; kc kd]。该运算对矩阵加法满足分配律：k(A + B) = kA + kB。

Negative scalar multiplication works as expected: −A simply changes the sign of each element. This can be combined with subtraction: A − B = A + (−B).

负标量乘法同样适用：−A 只是改变每个元素的符号。这可以结合减法使用：A − B = A + (−B)。

5. Matrix Multiplication | 矩阵乘法

Matrix multiplication is not simply multiplying corresponding elements. It involves a row-by-column process. You can multiply matrix A by matrix B only if the number of columns in A equals the number of rows in B.

矩阵乘法并不是简单地将对应元素相乘，而是采用行乘以列的方式。只有当 矩阵 A 的列数等于矩阵 B 的行数 时，你才能将 A 乘以 B。

If A is of order m × n and B is of order n × p, the product AB will have order m × p. To find the element in row i, column j of AB, multiply each element in row i of A by the corresponding element in column j of B, and sum the results.

如果 A 的阶是 m × n，B 的阶是 n × p，那么乘积 AB 的阶将是 m × p。要找到 AB 中第 i 行第 j 列的元素，需要将 A 的第 i 行中的每个元素与 B 的第 j 列中的对应元素相乘，然后将结果相加。

For 2 × 2 matrices: if A = [a b; c d] and B = [p q; r s], then AB = [ap+br aq+bs; cp+dr cq+ds]. Notice that matrix multiplication is not commutative: in general, AB ≠ BA.

对于 2 × 2 矩阵：如果 A = [a b; c d] 且 B = [p q; r s]，那么 AB = [ap+br aq+bs; cp+dr cq+ds]。注意矩阵乘法 不满足交换律：通常 AB ≠ BA。

Associativity holds: (AB)C = A(BC). Distributive laws: A(B + C) = AB + AC and (A + B)C = AC + BC.

结合律成立：(AB)C = A(BC)。分配律成立：A(B + C) = AB + AC 和 (A + B)C = AC + BC。

6. Identity and Zero Matrices | 单位矩阵与零矩阵

The identity matrix, denoted by I, acts like the number 1 in matrix multiplication. For any square matrix A of order n × n, AI = IA = A. The 2 × 2 identity matrix is I = [1 0; 0 1].

单位矩阵，记作 I，在矩阵乘法中的作用类似于数字 1。对于任何 n × n 的方阵 A，都有 AI = IA = A。2 × 2 的单位矩阵是 I = [1 0; 0 1]。

The zero matrix, denoted by O, has all elements equal to zero. Adding the zero matrix does not change a matrix: A + O = A. Multiplying by a zero matrix gives a zero matrix: A × O = O and O × A = O, provided the orders are compatible.

零矩阵，记作 O，所有元素均为零。加上零矩阵不会改变原矩阵：A + O = A。乘以零矩阵则得到零矩阵：A × O = O 且 O × A = O，前提是阶兼容。

These matrices are important for solving equations and proving properties. Remember that a zero determinant indicates the matrix does not have an inverse, much like division by zero.

这些矩阵对于求解方程和证明性质非常重要。请记住，行列式为零意味着该矩阵没有逆矩阵，就像除以零一样。

7. Determinant of a 2×2 Matrix | 2×2 矩阵的行列式

The determinant is a special number calculated from a square matrix. For a 2 × 2 matrix A = [a b; c d], the determinant is given by det(A) or |A|, where:

行列式是由方阵计算得出的一个特殊数值。对于 2 × 2 矩阵 A = [a b; c d]，其行列式记作 det(A) 或 |A|，计算公式为：

det(A) = ad − bc

If det(A) = 0, the matrix is called singular, meaning it has no inverse. If det(A) ≠ 0, the matrix is non-singular and an inverse exists. Determinants also help in understanding area scale factors in transformations.

如果 det(A) = 0，该矩阵被称为 奇异矩阵，这意味着它不存在逆矩阵。如果 det(A) ≠ 0，该矩阵是 非奇异的，逆矩阵存在。行列式还有助于理解几何变换中的面积缩放因子。

Example: For A = [4 2; 1 3], det(A) = (4)(3) − (2)(1) = 12 − 2 = 10.

示例：对于 A = [4 2; 1 3]，det(A) = (4)(3) − (2)(1) = 12 − 2 = 10。

8. Inverse of a 2×2 Matrix | 2×2 矩阵的逆矩阵

The inverse of a square matrix A, written A⁻¹, is the matrix such that A × A⁻¹ = I. Only non-singular matrices have inverses. For a 2 × 2 matrix A = [a b; c d], the inverse is:

方阵 A 的逆矩阵，记作 A⁻¹，满足 A × A⁻¹ = I。只有非奇异矩阵才有逆矩阵。对于 2 × 2 矩阵 A = [a b; c d]，其逆矩阵为：

A⁻¹ = (1 / det(A)) × [d −b; −c a]

You swap the positions of a and d, change the signs of b and c, and multiply by the reciprocal of the determinant. Always check that det(A) ≠ 0 before calculating the inverse.

你需要交换 a 和 d 的位置，改变 b 和 c 的符号，然后乘以行列式的倒数。在计算逆矩阵之前，务必先检查 det(A) ≠ 0。

Example: B = [5 2; 2 1]; det(B) = (5)(1) − (2)(2) = 1. Then B⁻¹ = (1/1) × [1 −2; −2 5] = [1 −2; −2 5].

示例：B = [5 2; 2 1]；det(B) = (5)(1) − (2)(2) = 1。因此 B⁻¹ = (1/1) × [1 −2; −2 5] = [1 −2; −2 5]。

9. Solving Simultaneous Equations | 解联立方程组

Matrices provide an elegant way to solve a pair of linear simultaneous equations. Write the equations in matrix form AX = B, where A is the coefficient matrix, X is the column matrix of variables, and B is the column matrix of constants.

矩阵为求解二元一次联立方程组提供了一种简洁的方法。将方程组写成矩阵形式 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。

For example, the system 2x + 3y = 5 and 4x + y = 6 can be written as:

例如，方程组 2x + 3y = 5 和 4x + y = 6 可以写成：

[2 3; 4 1] [x; y] = [5; 6]

If A is non-singular, multiply both sides on the left by A⁻¹ to obtain X = A⁻¹B. This gives the values of x and y directly. The order matters: A⁻¹ must be on the left of B.

如果 A 是非奇异的，两边同时左乘 A⁻¹，可得 X = A⁻¹B。这直接给出了 x 和 y 的值。注意次序：A⁻¹ 必须在 B 的左边。

Let us solve the above system: det(A) = (2)(1) − (3)(4) = 2 − 12 = −10. A⁻¹ = (1/−10) × [1 −3; −4 2] = [−0.1 0.3; 0.4 −0.2]. Then [x; y] = A⁻¹[5; 6] = [−0.1×5 + 0.3×6; 0.4×5 + (−0.2)×6] = [−0.5 + 1.8; 2.0 − 1.2] = [1.3; 0.8]. So x = 1.3, y = 0.8.

我们来解上述方程组：det(A) = (2)(1) − (3)(4) = −10。A⁻¹ = (1/−10) × [1 −3; −4 2] = [−0.1 0.3; 0.4 −0.2]。然后 [x; y] = A⁻¹[5; 6] = [−0.1×5 + 0.3×6; 0.4×5 + (−0.2)×6] = [1.3; 0.8]。因此 x = 1.3, y = 0.8。

10. Transformation Matrices | 变换矩阵

A 2 × 2 matrix can represent a geometric transformation of points in the plane. The column vectors of the matrix are the images of the unit vectors (1, 0) and (0, 1). This makes it easy to find the matrix for a given transformation or to identify the transformation from a matrix.

一个 2 × 2 矩阵可以表示平面上点的几何变换。矩阵的列向量就是单位向量 (1, 0) 和 (0, 1) 的像。这使得根据变换找出矩阵，或根据矩阵识别变换变得十分方便。

Common transformation matrices you should know include:

你需要掌握的常见变换矩阵有：

Transformation	Matrix	Description
Reflection in x-axis	[1 0; 0 −1]	(x, y) → (x, −y)
Reflection in y-axis	[−1 0; 0 1]	(x, y) → (−x, y)
Reflection in line y = x	[0 1; 1 0]	(x, y) → (y, x)
Reflection in line y = −x	[0 −1; −1 0]	(x, y) → (−y, −x)
Rotation 90° anticlockwise about O	[0 −1; 1 0]	(x, y) → (−y, x)
Rotation 180° about O	[−1 0; 0 −1]	(x, y) → (−x, −y)
Enlargement, scale factor k, centre O	[k 0; 0 k]	(x, y) → (kx, ky)
Stretch parallel to x-axis, factor k	[k 0; 0 1]	(x, y) → (kx, y)

The determinant of a transformation matrix gives the area scale factor. If det = 0, the transformation collapses the shape onto a line or point.

变换矩阵的行列式表示面积缩放因子。如果行列式为 0，则该变换会将图形压缩到一条直线或一个点上。

11. Common Mistakes and Exam Tips | 常见错误与应试技巧

Avoid these frequent errors to boost your marks:

避免以下常见错误，提升你的分数：

Attempting to add or subtract matrices of different orders. Always check orders first.

试图将不同阶的矩阵相加或相减。务必先检查阶是否一致。
Assuming matrix multiplication is commutative. AB usually does not equal BA, so keep the order as given.

误以为矩阵乘法满足交换律。AB 通常不等于 BA，因此保持题目给定的乘法顺序。
Forgetting to check that the determinant is non-zero before finding an inverse. If det = 0, state that no inverse exists.

在求逆矩阵之前忘记检查行列式是否非零。如果 det = 0，应说明逆矩阵不存在。
Writing the inverse formula incorrectly: remember it is (1/det) × [d −b; −c a], not [a b; c d] with swapped diagonals.

错误地书写逆矩阵公式：请牢记公式是 (1/det) × [d −b; −c a]，而不是简单交换对角线元素。
Mixing up the order when multiplying by the inverse to solve equations. Use X = A⁻¹B, not BA⁻¹.

在乘以逆矩阵解方程时弄错次序。应使用 X = A⁻¹B，而非 BA⁻¹。
Confusing the image of (1,0) with the first column and (0,1) with the second column in transformation matrices. Sketching the images can help.

在变换矩阵中，混淆 (1,0) 的像（对应第一列）和 (0,1) 的像（对应第二列）。画草图可以帮助理清思路。
When multiplying matrices by hand, organise your working neatly to avoid arithmetic slips. Write the row and column you are combining.

手动进行矩阵乘法时，要整齐地组织计算过程，避免算术错误。可以写出你所组合的行和列。

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GCSE CIE Maths: Matrices Explained | GCSE CIE 数学：矩阵考点精讲