IGCSE Edexcel Maths: Matrices Revision Guide | IGCSE Edexcel 数学：矩阵考点精讲

📚 IGCSE Edexcel Maths: Matrices Revision Guide | IGCSE Edexcel 数学：矩阵考点精讲

Matrices form an essential topic in IGCSE Edexcel Mathematics. They are powerful tools for organising data, solving systems of linear equations, and performing geometric transformations. This guide covers all the key concepts you need, from basic operations to finding inverses and using matrices for transformations. Whether you are preparing for Paper 2 or Paper 4, a solid understanding of matrices will boost your confidence and your score.

矩阵是 IGCSE Edexcel 数学中的重要内容。它们是整理数据、求解线性方程组以及执行几何变换的有力工具。本指南涵盖所有你需要的关键概念，从基本运算到求逆矩阵以及用矩阵进行变换。无论你是在准备 Paper 2 还是 Paper 4，扎实掌握矩阵都能提升你的信心与成绩。

1. What is a Matrix? | 什么是矩阵？

A matrix is a rectangular array of numbers, arranged in rows and columns. Each number inside a matrix is called an element or entry. Matrices are usually denoted by capital letters, for example A, B, or M. They are useful for storing information and performing calculations in a structured way.

矩阵是一个由行和列排列的数字阵列。矩阵中的每个数字称为元素。矩阵通常用大写字母表示，例如 A、B 或 M。它们有助于以结构化的方式储存信息并进行计算。

For instance, matrix A = [ 2 3; 1 5 ] has 2 rows and 2 columns. This is written as a 2 × 2 matrix.

例如，矩阵 A = [ 2 3; 1 5 ] 有 2 行和 2 列。这记作 2 × 2 矩阵。

2. Order of a Matrix | 矩阵的阶

The order (or dimension) of a matrix is given by the number of rows followed by the number of columns. We write it as m × n, where m is the number of rows and n is the number of columns. An m × n matrix is read as ‘m by n’.

矩阵的阶（或维数）由行数和列数给出。我们写作 m × n，其中 m 是行数，n 是列数。m × n 矩阵读作“m 乘 n”。

For example, a matrix with 3 rows and 2 columns has order 3 × 2. In IGCSE, you will mainly work with 2 × 2 matrices, but you may also see 2 × 1 or 1 × 2 matrices.

例如，拥有 3 行 2 列的矩阵的阶为 3 × 2。在 IGCSE 中，你主要处理 2 × 2 矩阵，但也可能见到 2 × 1 或 1 × 2 矩阵。

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

You can only add or subtract matrices if they have exactly the same order. To do this, simply add or subtract the corresponding elements. The operation is straightforward and follows the rules of ordinary arithmetic.

只有当矩阵阶数完全相同时，才能对它们进行加法或减法运算。只需将对应元素相加或相减。运算直接明了，遵循普通算术规则。

If A = [ a b; c d ] and B = [ e f; g h ], then A + B = [ a+e b+f; c+g d+h ]. Similarly, subtraction works element-wise.

若 A = [ a b; c d ] 且 B = [ e f; g h ]，则 A + B = [ a+e b+f; c+g d+h ]。减法同理按元素进行。

Watch out for common mistakes: always check that the orders match. A 2 × 2 matrix cannot be added to a 2 × 1 matrix.

注意常见错误：务必检查阶数是否一致。一个 2 × 2 矩阵不能与 2 × 1 矩阵相加。

4. Scalar Multiplication | 标量乘法

Scalar multiplication involves multiplying every element of a matrix by a constant number (called a scalar). If k is a scalar and M is a matrix, then kM is obtained by multiplying each entry of M by k. This operation always produces a matrix of the same order as M.

标量乘法是将矩阵的每个元素乘以一个常数（称为标量）。如果 k 是标量，M 是矩阵，那么 kM 就是将 M 的每个元素乘以 k。该运算总是产生与 M 同阶的矩阵。

For example, if M = [ 4 -1; 0 3 ] and k = 2, then 2M = [ 8 -2; 0 6 ].

例如，若 M = [ 4 -1; 0 3 ] 且 k = 2，则 2M = [ 8 -2; 0 6 ]。

5. Matrix Multiplication | 矩阵乘法

Matrix multiplication is not as simple as addition. You can multiply two matrices only when the number of columns in the first matrix equals the number of rows in the second matrix. The resulting matrix will have the number of rows of the first and the number of columns of the second.

矩阵乘法不像加法那么简单。只有当第一个矩阵的列数等于第二个矩阵的行数时，你才能将两者相乘。结果矩阵的行数等于第一个矩阵的行数，列数等于第二个矩阵的列数。

For 2 × 2 matrices, if A = [ a b; c d ] and B = [ e f; g h ], then AB = [ ae+bg af+bh; ce+dg cf+dh ]. Notice how you multiply rows of A by columns of B and sum the products.

对于 2 × 2 矩阵，若 A = [ a b; c d ] 且 B = [ e f; g h ]，则 AB = [ ae+bg af+bh; ce+dg cf+dh ]。注意你用 A 的行乘以 B 的列并将乘积求和。

A useful exam tip: use your fingers to track the row and column you are working on. This reduces errors in the multiplication order.

一个有用的考试技巧：用手指跟踪你正在处理的行和列。这能减少乘法顺序上的错误。

6. Properties of Matrix Multiplication | 矩阵乘法的性质

Matrix multiplication is not commutative in general. This means that for two matrices A and B, AB is often not equal to BA. In some special cases they might be equal, but you should never assume commutativity. Always multiply in the order given.

矩阵乘法通常不满足交换律。这意味着对于两个矩阵 A 和 B，AB 往往不等于 BA。在某些特殊情况下它们可能相等，但你绝对不能假设可交换。始终按照给定的顺序相乘。

However, matrix multiplication is associative: (AB)C = A(BC). It is also distributive: A(B + C) = AB + AC, provided the orders are compatible.

然而，矩阵乘法满足结合律：(AB)C = A(BC)。它也满足分配律：A(B + C) = AB + AC，前提是阶数兼容。

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

The identity matrix, denoted by I, is a special square matrix. For a 2 × 2 matrix, I = [ 1 0; 0 1 ]. Multiplying any matrix by the identity matrix (of compatible order) leaves it unchanged: AI = IA = A. It acts like the number 1 in ordinary multiplication.

单位矩阵记作 I，是一个特殊的方阵。对于 2 × 2 矩阵，I = [ 1 0; 0 1 ]。将任何矩阵乘以单位矩阵（阶数兼容）会使其保持不变：AI = IA = A。它就像普通乘法中的数字 1。

The zero matrix, often denoted by O, has all entries equal to zero. Adding the zero matrix to any matrix does not change that matrix. Multiplying by a zero matrix (where order allows) results in a zero matrix.

零矩阵通常记作 O，所有元素均为零。将零矩阵加到任何矩阵上都不会改变该矩阵。乘以零矩阵（阶数允许时）会得到零矩阵。

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

For a 2 × 2 matrix M = [ a b; c d ], the determinant is given by det(M) = ad − bc. This is a single number that tells us very important information, such as whether the matrix has an inverse. If the determinant is zero, the matrix is called singular and does not have an inverse.

对于 2 × 2 矩阵 M = [ a b; c d ]，行列式为 det(M) = ad − bc。这个单一数值能告诉我们非常重要的信息，例如矩阵是否有逆。如果行列式为零，则矩阵称为奇异矩阵，没有逆矩阵。

det(M) = ad − bc

Calculating the determinant correctly is crucial. Watch out for negative signs; mistakes in sign are a common source of lost marks.

正确计算行列式至关重要。注意负号；符号错误是常见的失分原因。

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

The inverse of a matrix M, denoted M⁻¹, is the matrix such that MM⁻¹ = M⁻¹M = I. Not every matrix has an inverse. For a 2 × 2 matrix M = [ a b; c d ], the inverse exists only if det(M) ≠ 0, and is given by:

矩阵 M 的逆，记作 M⁻¹，是满足 MM⁻¹ = M⁻¹M = I 的矩阵。并非每个矩阵都有逆。对于 2 × 2 矩阵 M = [ a b; c d ]，仅当 det(M) ≠ 0 时逆存在，并由下式给出：

M⁻¹ = (1 / det(M)) × [ d −b; −c a ]

Notice that the positions of a and d swap, and the signs of b and c change. Always check your answer by multiplying M by M⁻¹; you should get the identity matrix.

注意 a 和 d 的位置交换，b 和 c 的符号改变。务必通过将 M 乘以 M⁻¹ 来检查答案；你应该得到单位矩阵。

10. Solving Simultaneous Equations Using Matrices | 使用矩阵解联立方程组

Matrices provide an elegant method for solving a pair of linear simultaneous equations. Write the equations in the form ax + by = e and cx + dy = f. Then represent them as a matrix equation: M [x; y] = [e; f], where M = [ a b; c d ].

矩阵为求解一对线性联立方程提供了一种简洁的方法。将方程写成 ax + by = e 和 cx + dy = f 的形式。然后以矩阵方程表示：M [x; y] = [e; f]，其中 M = [ a b; c d ]。

To solve for x and y, multiply both sides by the inverse of M: [x; y] = M⁻¹ [e; f], provided M⁻¹ exists. This gives a quick algebraic solution without elimination or substitution.

要求解 x 和 y，将两边乘以 M 的逆：[x; y] = M⁻¹ [e; f]，前提是 M⁻¹ 存在。这无需消元或代入，即可快速给出代数解。

Be careful: if det(M) = 0, the system either has no unique solution or no solution at all. You may need to interpret this result in the exam.

注意：如果 det(M) = 0，该方程组要么没有唯一解，要么根本没有解。你可能需要在考试中解释这一结果。

11. Transformation Matrices | 变换矩阵

Matrices can represent geometric transformations in the coordinate plane. A 2 × 2 matrix can be applied to position vectors to produce an image. Common transformation matrices include:

矩阵可以表示坐标平面中的几何变换。一个 2 × 2 矩阵可作用于位置向量以产生像。常见的变换矩阵包括：

Rotation 90° anticlockwise about the origin: [ 0 −1; 1 0 ]

绕原点逆时针旋转 90°：[ 0 −1; 1 0 ]
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 ]
Enlargement scale factor k, centre origin: [ k 0; 0 k ]

以原点为中心、缩放因子为 k 的放大：[ k 0; 0 k ]

To find the image of a point, write it as a column vector and multiply the transformation matrix by that vector. Understanding these geometric links often appears in the Edexcel exam.

要找到点的像，将其写作列向量并用变换矩阵乘以该向量。理解这些几何联系在 Edexcel 考试中经常出现。

12. Key Exam Tips and Common Errors | 关键考试提醒与常见错误

Top tips for matrices in IGCSE: always check matrix orders before adding or multiplying. Write out each step clearly. For inverses, remember the formula: swap a and d, change signs of b and c, then divide by the determinant.

IGCSE 矩阵考试的最佳技巧：加或乘之前务必检查矩阵的阶。清晰地写出每一步。对于逆矩阵，记住公式：交换 a 和 d，改变 b 和 c 的符号，然后除以行列式。

Common mistakes include forgetting that matrix multiplication is not commutative, miscalculating determinants (especially with negative numbers), and trying to add matrices of different sizes. Practise regularly with past paper questions to build speed and accuracy.

常见错误包括忘记矩阵乘法不满足交换律、行列式计算错误（尤其是涉及负数时），以及试图将不同大小的矩阵相加。定期用历年真题练习，以提升速度与准确性。

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IGCSE Edexcel Maths: Matrices Revision Guide | IGCSE Edexcel 数学：矩阵 考点精讲