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

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

Matrices are a fundamental topic in GCSE Mathematics, particularly for students taking the Extended syllabus or additional maths qualifications. A matrix is a rectangular array of numbers arranged in rows and columns, and it provides a powerful way to organise data, perform transformations, and solve systems of equations. This revision guide covers all the key concepts you need to master for your exam, including matrix operations, determinants, inverses, and their applications. Work through each section carefully, practise the examples, and you will build the confidence to tackle any matrix question.

矩阵是 GCSE 数学中的一个核心主题，尤其对参加扩展课程或附加数学考试的学生来说尤为重要。矩阵是按行和列排列的数字矩形阵列，它提供了一种组织数据、进行变换和求解方程组的强大方法。本考点精讲涵盖了你需要掌握的矩阵运算、行列式、逆矩阵及其应用等所有关键概念。认真学完每个部分，练习例题，你将能自信应对任何矩阵考题。

1. What is a Matrix? | 什么是矩阵？

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. Each individual number inside a matrix is called an element or an entry. The dimension or order of a matrix is described by the number of rows followed by the number of columns. For example, a matrix with 2 rows and 3 columns is called a ‘2 by 3’ matrix, written as 2 × 3. Matrices are usually denoted by capital letters such as A, B, C.

矩阵是一个按行和列排列的数字、符号或表达式的矩形阵列。矩阵内的每个单独数字称为元素或项。矩阵的维数或阶数用行数和列数来描述，先写行数后写列数。例如，一个具有 2 行 3 列的矩阵被称为“2 乘 3”矩阵，记作 2 × 3。矩阵通常用大写字母表示，如 A、B、C。

A general matrix A of order m × n can be written as:

一个 m × n 阶的一般矩阵 A 可以表示为：

A = [aᵢⱼ] with i = 1, 2, …, m and j = 1, 2, …, n

Here aᵢⱼ represents the element in the i-th row and j-th column. A matrix with only one row is called a row matrix, and a matrix with only one column is a column matrix. A matrix with the same number of rows and columns is called a square matrix.

这里 aᵢⱼ 表示第 i 行第 j 列的元素。只有一行的矩阵称为行矩阵，只有一列的矩阵称为列矩阵。行数和列数相等的矩阵称为方阵。

2. Matrix Addition and Subtraction | 矩阵的加法和减法

Two matrices can only be added or subtracted if they have the exact same order. To add or subtract matrices, you simply add or subtract the corresponding elements in each position. The result will be a new matrix of the same order. This operation is both commutative and associative for addition, meaning A + B = B + A and (A + B) + C = A + (B + C).

只有当两个矩阵的阶数完全相同时，它们才能相加或相减。要进行矩阵加减，只需将每个位置上对应的元素相加或相减。结果将是一个同阶的新矩阵。矩阵加法满足交换律和结合律，即 A + B = B + A，(A + B) + C = A + (B + C)。

Example: If A = [2 3; 1 4] and B = [5 1; 2 0], then A + B = [2+5 3+1; 1+2 4+0] = [7 4; 3 4]. For subtraction, A − B = [2−5 3−1; 1−2 4−0] = [−3 2; −1 4].

示例：若 A = [2 3; 1 4]，B = [5 1; 2 0]，则 A + B = [7 4; 3 4]。A − B = [−3 2; −1 4]。

A zero matrix, denoted by O, is a matrix where all elements are zero. Adding a zero matrix of the same order leaves any matrix unchanged: A + O = A. Subtraction of a matrix from itself gives the zero matrix: A − A = O.

零矩阵（记作 O）是所有元素均为零的矩阵。加一个同阶的零矩阵不会改变原矩阵：A + O = A。矩阵减去自身得到零矩阵：A − A = O。

Remember that matrix addition and subtraction are element-wise operations, so you must always check that the matrices have matching dimensions before performing the calculation. A common mistake is trying to add a 2 × 3 matrix to a 3 × 2 matrix — this is not allowed.

请记住，矩阵加减是按元素进行的运算，因此在计算前必须检查矩阵的维数是否匹配。一个常见错误是试图将一个 2 × 3 矩阵与一个 3 × 2 矩阵相加——这是不允许的。

3. Scalar Multiplication | 标量乘法

Scalar multiplication refers to multiplying every element of a matrix by a single number (called a scalar). If k is a scalar and A is a matrix, the product kA is obtained by multiplying each element of A by k. The order of the matrix remains unchanged. This operation is very useful when combining transformations or simplifying expressions involving matrices.

标量乘法是指将矩阵的每个元素乘以一个单独的数（称为标量）。如果 k 是一个标量，A 是一个矩阵，则乘积 kA 通过将 A 的每个元素乘以 k 得到。矩阵的阶数不变。在组合变换或化简含有矩阵的表达式时，这种运算非常有用。

For example, if B = [1 −2; 0 3] and the scalar k = 4, then 4B = [4×1 4×(−2); 4×0 4×3] = [4 −8; 0 12]. Negative scalars simply change the sign of each element: −1 × B = [−1 2; 0 −3].

例如，若 B = [1 −2; 0 3]，标量 k = 4，则 4B = [4 −8; 0 12]。负标量则改变每个元素的符号：−1 × B = [−1 2; 0 −3]。

Scalar multiplication is distributive over matrix addition: k(A + B) = kA + kB, and also (k + l)A = kA + lA for scalars k and l. These properties allow you to simplify matrix expressions algebraically just as you would with ordinary numbers.

标量乘法对矩阵加法满足分配律：k(A + B) = kA + kB，且对于标量 k 和 l 有 (k + l)A = kA + lA。这些性质使你能够像处理普通数字一样对矩阵表达式进行代数化简。

4. Matrix Multiplication | 矩阵乘法

Matrix multiplication is not element-wise — it involves a row-by-column operation. To multiply two matrices A and B, the number of columns in A must equal the number of rows in B. If A is of order m × n and B is of order n × p, then the product AB will be an m × p matrix. The element in the i-th row and j-th column of AB is calculated by taking the dot product of the i-th row of A with the j-th column of B.

矩阵乘法不是按元素进行的——它涉及行与列的运算。要计算两个矩阵 A 和 B 的乘积，A 的列数必须等于 B 的行数。如果 A 是 m × n 阶，B 是 n × p 阶，则乘积 AB 将是一个 m × p 阶矩阵。AB 中第 i 行第 j 列的元素是通过将 A 的第 i 行与 B 的第 j 列做点积计算得到的。

For matrices A = [a b; c d] and B = [e f; g h], the product AB is:

对于矩阵 A = [a b; c d] 和 B = [e f; g h]，乘积 AB 为：

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

Example: Let A = [2 1; 3 0] and B = [1 4; 2 5]. Then AB = [2×1+1×2 2×4+1×5; 3×1+0×2 3×4+0×5] = [4 13; 3 12]. Notice that BA would be different: BA = [1×2+4×3 1×1+4×0; 2×2+5×3 2×1+5×0] = [14 1; 19 2]. This demonstrates that matrix multiplication is not commutative, i.e., AB ≠ BA in general.

示例：设 A = [2 1; 3 0]，B = [1 4; 2 5]。则 AB = [4 13; 3 12]。注意 BA 不同：BA = [14 1; 19 2]。这说明矩阵乘法不满足交换律，即通常 AB ≠ BA。

Matrix multiplication is associative: (AB)C = A(BC), and distributive: A(B + C) = AB + AC. When multiplying, be careful with the order — ‘pre-multiply’ and ‘post-multiply’ produce different results unless the matrices are inverses of each other or special cases like the identity matrix.

矩阵乘法满足结合律：(AB)C = A(BC)，以及分配律：A(B + C) = AB + AC。乘法时需注意顺序——“左乘”和“右乘”会产生不同的结果，除非矩阵互为逆矩阵或特殊矩阵（如单位矩阵）。

5. The Identity Matrix | 单位矩阵

The identity matrix, denoted by I, is a special square matrix that acts like the number 1 in ordinary multiplication. For any square matrix A of the same order, AI = IA = A. The identity matrix has 1s on the main diagonal (from top left to bottom right) and 0s everywhere else. The 2 × 2 identity matrix is I₂ = [1 0; 0 1], and the 3 × 3 identity matrix is I₃ = [1 0 0; 0 1 0; 0 0 1].

单位矩阵（记作 I）是一种特殊的方阵，其作用类似于普通乘法中的数字 1。对于任何同阶方阵 A，都有 AI = IA = A。单位矩阵的主对角线（从左上到右下）上元素为 1，其余位置为 0。2 × 2 单位矩阵为 I₂ = [1 0; 0 1]，3 × 3 单位矩阵为 I₃ = [1 0 0; 0 1 0; 0 0 1]。

If you multiply any matrix by the identity matrix (provided the dimensions allow it), you get the original matrix back. This property is vital when discussing inverse matrices, because the inverse of a matrix A is the matrix A⁻¹ such that AA⁻¹ = A⁻¹A = I. The identity matrix is also the neutral element for matrix multiplication.

任何矩阵乘以单位矩阵（只要维度允许）都会得到原矩阵。这一性质在讨论逆矩阵时至关重要，因为矩阵 A 的逆 A⁻¹ 满足 AA⁻¹ = A⁻¹A = I。单位矩阵也是矩阵乘法的中性元素。

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

The determinant is a special number that can be calculated from a square matrix. It tells us whether a matrix has an inverse (if the determinant is non-zero) or not (if determinant is zero). For a 2 × 2 matrix A = [a b; c d], the determinant, denoted det(A) or |A|, is given by:

行列式是可以从方阵中计算出的一个特殊数。它告诉我们一个矩阵是否有逆矩阵（行列式非零时有逆，为零时没有）。对于 2 × 2 矩阵 A = [a b; c d]，行列式记作 det(A) 或 |A|，计算公式为：

det(A) = ad − bc

For example, if A = [3 2; 1 4], then det(A) = (3 × 4) − (2 × 1) = 12 − 2 = 10. Since det(A) ≠ 0, this matrix has an inverse. If A = [2 6; 1 3], then det(A) = (2×3) − (6×1) = 6 − 6 = 0, so this matrix does not have an inverse — it is called a singular matrix.

例如，若 A = [3 2; 1 4]，则 det(A) = 10 ≠ 0，该矩阵有逆矩阵。若 A = [2 6; 1 3]，则 det(A) = 0，该矩阵没有逆矩阵——称为奇异矩阵。

Determinants also have geometric meaning: the absolute value of the determinant of a 2 × 2 transformation matrix gives the area scale factor of the transformation. A negative determinant indicates that the transformation involves a reflection.

行列式还有几何意义：2 × 2 变换矩阵行列式的绝对值表示该变换的面积比例因子。负的行列式表示变换包含反射。

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

The inverse of a 2 × 2 matrix A, denoted A⁻¹, is the matrix such that AA⁻¹ = A⁻¹A = I. It exists only if det(A) ≠ 0. For A = [a b; c d], the formula for the inverse is:

2 × 2 矩阵 A 的逆矩阵记作 A⁻¹，满足 AA⁻¹ = A⁻¹A = I。它仅在 det(A) ≠ 0 时存在。对于 A = [a b; c d]，逆矩阵的公式为：

A⁻¹ = 1 / det(A) × [d −b; −c a]

Notice the steps: swap the elements a and d, change the signs of b and c, then multiply the resulting matrix by 1/det(A). For example, if A = [4 7; 2 6], first det(A) = 4×6 − 7×2 = 24 − 14 = 10. Then A⁻¹ = 1/10 × [6 −7; −2 4] = [0.6 −0.7; −0.2 0.4]. You should always verify your answer by checking that A × A⁻¹ = I.

注意步骤：交换元素 a 和 d，改变 b 和 c 的符号，然后将所得矩阵乘以 1/det(A)。例如，若 A = [4 7; 2 6]，det(A) = 10，则 A⁻¹ = 1/10 × [6 −7; −2 4] = [0.6 −0.7; −0.2 0.4]。应始终通过验证 A × A⁻¹ = I 来检查答案。

If det(A) = 0, the matrix is singular and has no inverse. The inverse is used to solve matrix equations. For example, if AX = B, then multiplying both sides on the left by A⁻¹ gives X = A⁻¹B. This is a very common type of exam question.

若 det(A) = 0，则该矩阵是奇异矩阵，没有逆矩阵。逆矩阵用于求解矩阵方程。例如，若 AX = B，则对等式两边左乘 A⁻¹ 得 X = A⁻¹B。这是非常常见的考试题型。

8. Solving Simultaneous Equations using Matrices | 用矩阵解联立方程组

Matrices provide an efficient method for solving a system of two linear equations in two unknowns. Given the equations:

矩阵为求解含两个未知数的两个线性方程组提供了一种高效的方法。给定方程组：

a₁x + b₁y = c₁
a₂x + b₂y = c₂

We can represent it in matrix form as AX = B, where A = [a₁ b₁; a₂ b₂], X = [x; y], and B = [c₁; c₂]. The solution is X = A⁻¹B, provided det(A) ≠ 0. This means you compute the inverse of the coefficient matrix A, then multiply it by the constant matrix B to find the values of x and y.

我们可以将其表示为矩阵形式 AX = B，其中 A = [a₁ b₁; a₂ b₂]，X = [x; y]，B = [c₁; c₂]。解为 X = A⁻¹B，前提是 det(A) ≠ 0。也就是说，先求系数矩阵 A 的逆矩阵，再乘以常数矩阵 B，即可得到 x 和 y 的值。

Example: Solve 2x + 3y = 5 and x − y = 1. Write A = [2 3; 1 −1], B = [5; 1]. det(A) = 2(−1) − 3×1 = −2 − 3 = −5. So A⁻¹ = 1/−5 × [−1 −3; −1 2] = [0.2 0.6; 0.2 −0.4] (after simplifying). Then X = A⁻¹B = [0.2×5+0.6×1; 0.2×5−0.4×1] = [1+0.6; 1−0.4] = [1.6; 0.6]. Thus x = 1.6, y = 0.6. You can check by substituting back into the original equations.

示例：解方程组 2x + 3y = 5 和 x − y = 1。写出 A = [2 3; 1 −1]，B = [5; 1]。det(A) = −5，A⁻¹ = 1/−5 × [−1 −3; −1 2] = [0.2 0.6; 0.2 −0.4]。则 X = [1.6; 0.6]，故 x = 1.6, y = 0.6。可代入原方程组检验。

This method is particularly useful when you need to solve multiple systems with the same coefficient matrix but different constants, as you only need to compute the inverse once. However, remember that if det(A) = 0, the system either has no unique solution (it could be inconsistent or have infinitely many solutions).

当需要求解具有相同系数矩阵但常数项不同的多个方程组时，这种方法特别有用，因为你只需计算一次逆矩阵。但请记住，若 det(A) = 0，该方程组要么没有唯一解（可能矛盾，也可能有无穷多解）。

9. Transformations using Matrices | 用矩阵表示变换

In geometry, a 2 × 2 matrix can represent a transformation of points in the plane. When you multiply a position vector [x; y] by a transformation matrix, you get the new coordinates of the point after the transformation. Common transformations include reflections, rotations, enlargements, and shears. The first column of the transformation matrix is the image of the point (1, 0), and the second column is the image of (0, 1).

在几何中，2 × 2 矩阵可以表示平面内点的变换。当你用变换矩阵乘以位置向量 [x; y] 时，得到变换后点的新坐标。常见的变换包括反射、旋转、放大和剪切。变换矩阵的第一列是点 (1, 0) 的像，第二列是 (0, 1) 的像。

For example, the matrix for a reflection in the x-axis is [1 0; 0 −1], because (1, 0) stays at (1, 0) and (0, 1) goes to (0, −1). A rotation of 90° anticlockwise about the origin is represented by [0 −1; 1 0]. An enlargement by scale factor k is [k 0; 0 k].

例如，关于 x 轴反射的矩阵是 [1 0; 0 −1]，因为 (1, 0) 保持不变，(0, 1) 变为 (0, −1)。绕原点逆时针旋转 90° 的矩阵是 [0 −1; 1 0]。以比例因子 k 的放大用 [k 0; 0 k] 表示。

To find the image of a point P(x, y) under a transformation represented by matrix M, compute M × [x; y] as a column vector. If you have a shape with multiple vertices, you can combine the position vectors into a single matrix and multiply by M to get all transformed points at once. Also, combining transformations corresponds to multiplying their matrices: applying transformation A followed by B gives the matrix BA (note the reverse order).

要求点 P(x, y) 在矩阵 M 表示的变换下的像，计算 M × [x; y]。如果有一个多顶点的图形，可将位置向量合并成一个矩阵，再乘以 M 一次性得出所有变换后的点。另外，组合变换对应于矩阵相乘：先执行变换 A 再执行 B，得到的矩阵是 BA（注意逆序）。

10. Determinant and Area Scale Factor | 行列式与面积比例因子

When a transformation is represented by a 2 × 2 matrix M, the absolute value of the determinant |det(M)| gives the area scale factor of the transformation. This means if a shape has original area A, after transformation its area becomes |det(M)| × A. This is extremely useful for checking whether a transformation preserves area or for finding the area of an image without calculating all coordinates.

当变换由 2 × 2 矩阵 M 表示时，行列式的绝对值 |det(M)| 给出变换的面积比例因子。这意味着若原图形面积为 A，变换后其面积变为 |det(M)| × A。这对于检查变换是否保面积，或在未计算所有坐标的情况下求像的面积极为有用。

For example, a rotation matrix like [0 −1; 1 0] has determinant (0×0) − (−1×1) = 1, so rotation preserves area. A reflection matrix typically has determinant −1, but the absolute value is still 1, so reflections also preserve area. However, an enlargement by factor k has matrix [k 0; 0 k] and determinant k², so area is multiplied by k². A shear matrix may have determinant 1, meaning it preserves area despite changing shape.

例如，旋转矩阵 [0 −1; 1 0] 的行列式为 1，因此旋转保持面积不变。反射矩阵的行列式通常为 −1，但绝对值为 1，故反射也保面积。而放大因子为 k 的放大矩阵 [k 0; 0 k] 的行列式为 k²，因此面积乘以 k²。剪切矩阵的行列式可能为 1，这意味着尽管形状改变，面积保持不变。

11. Special Matrices and Their Properties | 特殊矩阵及其性质

Some matrices have special properties that help simplify calculations. A symmetric matrix is equal to its transpose: A = Aᵀ. The transpose is obtained by swapping rows and columns. A diagonal matrix has non-zero elements only on the main diagonal; multiplying by a diagonal matrix scales rows or columns very efficiently. An orthogonal matrix satisfies AᵀA = I, meaning its transpose is its inverse, common in rotation and reflection matrices.

某些矩阵具有特殊性质，有助于简化计算。对称矩阵等于其转置：A = Aᵀ。转置通过交换行与列得到。对角矩阵仅在主对角线上有非零元素；乘以对角矩阵可高效地缩放行或列。正交矩阵满足 AᵀA = I，即其转置就是其逆，常见于旋转和反射矩阵。

In GCSE, you are mainly expected to recognise the properties of the identity matrix and the zero matrix. The zero matrix, when multiplied by any conformable matrix, results in a zero matrix. You should also know that the determinant of a product equals the product of determinants: det(AB) = det(A)det(B). This can help find determinants of composite transformations.

在 GCSE 阶段，主要要求你识别单位矩阵和零矩阵的性质。零矩阵与任何可乘矩阵相乘都得到零矩阵。你还应知道乘积的行列式等于行列式的乘积：det(AB) = det(A)det(B)。这有助于求复合变换的行列式。

Also note that matrix multiplication is not commutative, but the identity matrix commutes with any square matrix. When working with inverses, remember that (AB)⁻¹ = B⁻¹A⁻¹ — the order reverses. This is a common trick in more challenging problems.

另请注意，矩阵乘法不满足交换律，但单位矩阵与任何方阵可交换。处理逆矩阵时，记住 (AB)⁻¹ = B⁻¹A⁻¹ ——顺序颠倒。这是难题中常见的技巧。

12. Common Errors and Exam Tips | 常见错误与考试技巧

Students often lose marks on matrix questions due to carelessness with order of multiplication, forgetting to check dimensions before multiplying, or miscalculating determinants. Always write the cross-multiplication steps for 2 × 2 determinants clearly. When finding the inverse, remember to calculate det(A) first; if it is zero, stop — the inverse does not exist. For transformation questions, be clear whether you are asked for the matrix that represents the transformation or the image of a specific point.

学生在矩阵题中常因马虎而丢分，例如乘法顺序写错、乘前忘记检查维数，或行列式计算错误。务必清晰写出 2 × 2 行列式的叉乘步骤。求逆矩阵时，记得先计算 det(A)；若为零，则停下——逆矩阵不存在。在变换题中，要看清问题是要求表示变换的矩阵，还是求具体点的像。

A top exam tip: after finding an inverse, multiply it by the original matrix as a check — you should get the identity matrix (allowing for small rounding if using decimals). When solving simultaneous equations, present the matrix method step by step: write the coefficient matrix, the variable matrix, the constant matrix, compute the inverse, and then multiply. This structured approach earns method marks even if the final answer has an error.

实用的考试技巧：求出逆矩阵后，将其乘以原矩阵进行校验——应得到单位矩阵（若使用小数允许微小舍入误差）。解联立方程组时，按步骤呈现矩阵方法：写出系数矩阵、变量矩阵、常数矩阵，计算逆矩阵，然后相乘。这种结构化的方法即使最终答案有误也能获得过程分。

Finally, practise past paper questions under timed conditions. Matrix questions are often straightforward if you follow the rules precisely, but they require accuracy. Memorise the inverse formula, determinant formula, and the conditions for matrix multiplication. Use your calculator efficiently where allowed, but always show your working to secure full marks.

最后，请在计时条件下练习往年真题。矩阵题如果严格按规则操作，往往很直接，但要求精确。记住逆矩阵公式、行列式公式以及矩阵乘法的条件。在允许使用计算器的情况下要高效运用，但一定要展示解题过程以确保得满分。

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GCSE Maths: Matrices Revision Guide | GCSE 数学：矩阵 考点精讲