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

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

Matrices are a fundamental part of the GCSE OCR Mathematics (Higher Tier / Further Maths) syllabus. Understanding how to work with matrices – from basic operations to solving linear equations and interpreting geometric transformations – will give you a strong advantage in your exam. This revision guide walks you through every key concept, with clear explanations and step‑by‑step examples.

矩阵是GCSE OCR数学（高阶/进阶数学）教学大纲的核心组成部分。理解矩阵的各种操作，从基本运算到解线性方程组以及理解几何变换，都会让你在考试中占据明显优势。本复习指南将带你梳理每一个关键概念，配以清晰的讲解和分步示例。

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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 entry is called an element. Matrices are usually denoted by capital letters, such as A, B or M. For example, a 2 × 2 matrix A can be represented as:

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。每个条目称为一个元素。矩阵通常用大写字母表示，如A、B或M。例如，一个2×2矩阵A可以表示为：

a	b
c	d

Here, a, b, c and d are the elements. The first row is (a b) and the second row is (c d). Matrices allow us to store and manipulate data efficiently, and they are especially useful for representing transformations and systems of equations.

这里a、b、c、d是矩阵的元素。第一行是(a b)，第二行是(c d)。矩阵让我们能够高效地存储和处理数据，对于表示变换和方程组尤为有用。

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2. Order of a Matrix | 矩阵的阶

The order (or size) of a matrix is given by ‘number of rows × number of columns’. A matrix with m rows and n columns is said to be of order m × n. For instance, the matrix below has 2 rows and 3 columns, so its order is 2 × 3.

矩阵的阶（或大小）由“行数 × 列数”给出。一个有m行n列的矩阵被称为m×n矩阵。例如，下面的矩阵有2行3列，因此它的阶是2×3。

1	0	−2
3	√2	5

When performing operations like addition or multiplication, the order of the matrices determines whether the operation is defined and what the resulting order will be. Always check the order first.

在进行加法或乘法等运算时，矩阵的阶决定了该运算是否成立以及结果矩阵的阶。一定要首先检查矩阵的阶。

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

You can add or subtract two matrices only if they have exactly the same order. The sum or difference is obtained by adding or subtracting the corresponding elements. For example, if A = [1 2; 3 4] and B = [5 6; 7 8], then A + B = [1+5 2+6; 3+7 4+8] = [6 8; 10 12].

只有当两个矩阵具有完全相同的阶时，你才能将它们相加或相减。和或差由对应元素相加或相减得到。例如，若A = [1 2; 3 4]，B = [5 6; 7 8]，则A + B = [1+5 2+6; 3+7 4+8] = [6 8; 10 12]。

Subtraction works in the same way: A − B = [1−5 2−6; 3−7 4−8] = [−4 −4; −4 −4]. Matrix addition is commutative (A + B = B + A) and associative ((A + B) + C = A + (B + C)).

减法规则相同：A − B = [1−5 2−6; 3−7 4−8] = [−4 −4; −4 −4]。矩阵加法满足交换律（A+B = B+A）和结合律（(A+B)+C = A+(B+C)）。

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4. Scalar Multiplication | 数乘

To multiply a matrix by a scalar (a single number), simply multiply every element of the matrix by that scalar. If k is a scalar and A is a matrix, then kA is the matrix where each element is k times the corresponding element of A.

要将一个矩阵乘以一个标量（一个单独的数字），只需用该标量乘以矩阵的每一个元素。若k是标量，A是矩阵，则kA是这样一个矩阵：它的每个元素都是A中对应元素的k倍。

Example: If A = [2 −1; 0 3] and k = 3, then 3A = [3×2 3×(−1); 3×0 3×3] = [6 −3; 0 9]. Scalar multiplication is commutative, so kA = Ak.

示例：若A = [2 −1; 0 3]，k = 3，则3A = [3×2 3×(−1); 3×0 3×3] = [6 −3; 0 9]。数乘满足交换律，即kA = Ak。

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5. Matrix Multiplication | 矩阵乘法

Matrix multiplication is not as straightforward as addition. Two matrices A and B can be multiplied only if the number of columns in A equals the number of rows in B. If A is m × n and B is n × p, then the product AB is an m × p matrix. The element in the i‑th row and j‑th column of AB is the dot product of the i‑th row of A and the j‑th column of B.

矩阵乘法不像加法那样简单。两个矩阵A和B可以相乘的条件是：A的列数等于B的行数。若A是m×n矩阵，B是n×p矩阵，则乘积AB是一个m×p矩阵。AB中第i行第j列的元素是A的第i行与B的第j列的点积。

For example, let A = [1 2; 3 4] and B = [2 0; 1 −1]. Then AB = [1×2+2×1 1×0+2×(−1); 3×2+4×1 3×0+4×(−1)] = [4 −2; 10 −4]. Notice that matrix multiplication is generally not commutative: AB ≠ BA. Always work systematically, row by column.

例如，令A = [1 2; 3 4]，B = [2 0; 1 −1]。那么AB = [1×2+2×1 1×0+2×(−1); 3×2+4×1 3×0+4×(−1)] = [4 −2; 10 −4]。请注意，矩阵乘法通常不满足交换律：AB ≠ BA。计算时要有条不紊，逐行乘逐列。

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6. The 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. For any 2×2 matrix A = [a b; c d], the determinant is given by:

行列式是可以从方阵计算出来的一个特殊数值。对于任意2×2矩阵A = [a b; c d]，行列式由下式给出：

det A = ad − bc

For example, if A = [3 5; 1 2], then det A = (3)(2) − (5)(1) = 6 − 5 = 1. The determinant tells us whether a matrix has an inverse: If det A = 0, the matrix is singular and has no inverse. If det A ≠ 0, the matrix is invertible.

例如，若A = [3 5; 1 2]，则det A = (3)(2) − (5)(1) = 6 − 5 = 1。行列式告诉我们矩阵是否有逆：若det A = 0，则矩阵奇异，没有逆矩阵；若det A ≠ 0，则矩阵可逆。

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

The inverse of a 2×2 matrix A, denoted A⁻¹, satisfies A A⁻¹ = A⁻¹ A = I, where I is the 2×2 identity matrix. For a matrix A = [a b; c d] with determinant Δ = ad − bc ≠ 0, the inverse is given by:

2×2矩阵A的逆矩阵记作A⁻¹，满足A A⁻¹ = A⁻¹ A = I，其中I是2×2单位矩阵。对于矩阵A = [a b; c d]，若其行列式Δ = ad − bc ≠ 0，则逆矩阵由下式给出：

A⁻¹ = 1/Δ × [ d −b ; −c a ]

To build A⁻¹, swap the positions of a and d, change the signs of b and c, then multiply each element by 1/Δ. Example: For A = [3 5; 1 2], Δ = 1, so A⁻¹ = [2 −5; −1 3]. You can check your answer by computing A A⁻¹ to see if you get the identity matrix.

构建A⁻¹的方法是：交换a和d的位置，将b和c变号，然后对每个元素乘以1/Δ。示例：对于A = [3 5; 1 2]，Δ = 1，因此A⁻¹ = [2 −5; −1 3]。你可以通过计算A A⁻¹来检验结果是否得到单位矩阵。

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8. Solving Linear Equations Using Matrices | 用矩阵解线性方程组

Matrices provide an efficient way to solve systems of two linear equations. Consider the equations:

矩阵提供了一种解两个线性方程组的高效方法。考虑方程组：

ax + by = e
cx + dy = f

This can be written in matrix form as AX = B, where:

这可以写成矩阵形式AX = B，其中：

a	b
c	d

is A, X = [x; y] and B = [e; f]. If det A ≠ 0, the solution is X = A⁻¹ B. Simply multiply the inverse matrix by the constant matrix B to find the values of x and y.

是A，X = [x; y]，B = [e; f]。如果det A ≠ 0，则解为X = A⁻¹ B。只需用逆矩阵乘以常数矩阵B，即可求出x和y的值。

For example, solve: 2x + 3y = 5, x − y = 1. Here A = [2 3; 1 −1], det A = (2)(−1) − (3)(1) = −2 − 3 = −5. Then A⁻¹ = 1/(−5) × [−1 −3; −1 2] = [1/5  3/5; 1/5  −2/5] (after multiplying). Multiply by B = [5; 1] to get X = [ (1/5)×5 + (3/5)×1 ; (1/5)×5 + (−2/5)×1 ] = [1 + 0.6; 1 − 0.4] = [1.6; 0.6]. So x = 1.6, y = 0.6.

例如，解方程组：2x + 3y = 5, x − y = 1。这里A = [2 3; 1 −1]，det A = (2)(−1) − (3)(1) = −2 − 3 = −5。那么A⁻¹ = 1/(−5) × [−1 −3; −1 2] = [1/5  3/5; 1/5  −2/5]（相乘后）。再乘以B = [5; 1]得X = [ (1/5)×5 + (3/5)×1 ; (1/5)×5 + (−2/5)×1 ] = [1 + 0.6; 1 − 0.4] = [1.6; 0.6]。因此x = 1.6，y = 0.6。

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

The 2×2 identity matrix, denoted I, is a special square matrix with 1s on the main diagonal and 0s elsewhere:

2×2单位矩阵记作I，是一个特殊的方阵，主对角线上是1，其余位置是0：

1	0
0	1

For any 2×2 matrix A, A I = I A = A. The identity matrix acts like the number 1 in ordinary multiplication. The 2×2 zero matrix, O, has all elements equal to 0. Adding the zero matrix to any matrix leaves it unchanged, and multiplying by the zero matrix gives the zero matrix.

对任意2×2矩阵A，有 A I = I A = A。

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