📚 Matrices for GCSE WJEC Mathematics: Exam Focus | GCSE WJEC 数学：矩阵 考点精讲
Matrices are a crucial topic in the WJEC GCSE Mathematics Higher Tier specification. They provide a powerful way to organise data and perform transformations, and mastering them can give you a significant advantage in the exam. This article covers all key concepts: from basic operations to using inverse matrices for solving simultaneous equations and applying transformation matrices for reflections, rotations and enlargements.
矩阵是 WJEC GCSE 数学高年级考试的一个重要主题。它们提供了一个组织数据和进行变换的强大工具，掌握矩阵知识能让你在考试中占得先机。本文涵盖所有关键概念：从基本运算到利用逆矩阵解联立方程，以及运用变换矩阵进行反射、旋转和放大。
1. What is a Matrix and Its Order? | 什么是矩阵及其阶？
rows and 3 columns has order 2 × 3. We usually denote matrices by bold capital letters, such as A, B, C.
矩阵是由数字排成的矩形阵列，按行和列排列。矩阵的阶写作 行数 × 列数。例如，一个有 2 行 3 列的矩阵，其阶为 2 × 3。我们通常用粗体大写字母表示矩阵，例如 A、B、C。
ed an element or entry. The element in the i-th row and j-th column is often written as aᵢⱼ. For example, A = [2 −1; 0 ts elements are a₁₁ = 2, a₁₂ = −1, a₂₁ = 0 and a₂₂ = 5. The semi-colon separates the rows.
矩阵中的每个数字称为元素。第 i 行第 j 列的元素常记为 aᵢⱼ。例如，A = [2 −1; 0 5] 是一个 2 × 2 矩阵，其元素为 a₁₁ = 2、a₁₂ = −1、a₂₁ = 0、a₂₂ = 5。分号用于分隔不同的行。
A = [2 −1; 0 5] order: 2 × 2
2. Matrix Addition and Subtraction | 矩阵的加减法
he same order. You then add or subtract the corresponding elements. For example, if A = [4 3; 2 1] and B = [1 0; −2 3], then A + B = [4+1 3+0; 2+(−2) 1+3] = [5 3; 0 4].
只有相同阶的矩阵才能相加或相减，然后直接对对应元素进行加减。例如，若 A = [4 3; 2 1] 且 B = [1 0; −2 3]，则 A + B = [4+1 3+0; 2+(−2) 1+3] = [5 3; 0 4]。
Subtraction works identically: A − B = [4−1 3−0; 2−(−2) 1−3] = [3 3; 4 −2]. If the orders are different, the operation is undefined in WJEC questions.
减法同理：A − B = [4−1 3−0; 2−(−2) 1−3] = [3 3; 4 −2]。如果两个矩阵的阶不同，该运算在 WJEC 考题中视为无定义。
A + B = [5 3; 0 4] A − B = [3 3; 4 −2]
3. Scalar Multiplication | 标量乘法
scalar (a single number), every element is multiplied by that scalar. If k = 3 and M = [2 −1; 0 4], then 3M = [3×2 3×(−1); 3×0 3×4] = [6 −3; 0 12].
当矩阵乘以一个标量（一个数）时，矩阵中的每一个元素都要乘以这个标量。若 k = 3 且 M = [2 −1; 0 4]，则 3M = [3×2 3×(−1); 3×0 3×4] = [6 −3; 0 12]。
Scalar multiplication is used in enlargement matrices and in simplifying matrix expressions. Always multiply every entry, not just the first row or first column.
标量乘法常用于放大矩阵以及化简矩阵表达式。务必乘以每一个元素，而不是只乘第一行或第一列。
3M = [6 −3; 0 12]
4. Matrix Multiplication | 矩阵乘法
× 2 matrices, multiply rows of the first matrix by columns of the second matrix. If A = [a b; c d] and B = [e f; g h], then AB = [ae+bg af+bh; ce+dg cf+dh].
矩阵乘法不是对应元素相乘。对于两个 2 × 2 矩阵，要用第一个矩阵的行乘以第二个矩阵的列。若 A = [a b; c d] 且 B = [e f; g h]，则 AB = [ae+bg af+bh; ce+dg cf+dh]。
need 2 × 2 by 2 × 2 multiplication and 2 × 2 by 2 × 1 column vectors for transformations.
乘法的顺序很重要：AB 通常不等于 BA。在 WJEC GCSE 高年级考试中，你主要需要掌握 2 × 2 乘 2 × 2 的乘法，以及 2 × 2 乘 2 × 1 列向量用于变换。
AB = [ae+bg af+bh; ce+dg cf+dh]
5. The Identity Matrix and Zero Matrix | 单位矩阵与零矩阵
The identity matrix I for 2 × 2 matrices is [1 0; 0 Multiplying any 2 × 2 matrix by I leaves it unchanged: AI = IA = A. The zero matrix O = [0 0; 0 0] contains all zeros and acts like 0 in matrix addition.
2 × 2 矩阵的单位矩阵 I 是 [1 0; 0 1]。任何 2 × 2 矩阵乘以 I 都保持不变：AI = IA = A。零矩阵 O = [0 0; 0 0] 的所有元素都是 0，在矩阵加法中类似于数字 0 的作用。
us to define the inverse of a matrix: A × A⁻¹ = A⁻¹ × A = I, provided the inverse exists.
单位矩阵很重要，因为它让我们可以定义矩阵的逆：A × A⁻¹ = A⁻¹ × A = I，前提是逆矩阵存在。
I = [1 0; 0 1] O = [0 0; 0 0]
6. Determinant of a 2 × 2 Matrix | 2 × 2 矩阵的行列式
The determinant of A = [a b; c d] is det(A) = ad − bc. For A = [2 −1; 0 t(A) = (2 × 5) − (−1 × 0) = 10 − 0 = 10. The determinant is a single number, not a matrix.
矩阵 A = [a b; c d] 的行列式是 det(A) = ad − bc。对于 A = [2 −1; 0 5]，det(A) = (2 × 5) − (−1 × 0) = 10 − 0 = 10。行列式是一个数，而不是一个矩阵。
gular and it has no inverse. WJEC questions often ask you to find the determinant before finding an inverse or solving simultaneous equations.
如果 det(A) = 0，该矩阵称为奇异矩阵，它没有逆矩阵。WJEC 题目常要求你先求行列式，再求逆矩阵或解联立方程。
det(A) = ad − bc
7. Inverse of a 2 × 2 Matrix | 2 × 2 矩阵的逆
The inverse of a 2 × 2 matrix A = [a b; c d] is A⁻¹ = (1/det(A)) × [d −b; −c t the numbers on the leading diagonal swap, and the signs of the other two entries change.
2 × 2 矩阵 A = [a b; c d] 的逆矩阵是 A⁻¹ = (1/det(A)) × [d −b; −c a]，前提是 det(A) ≠ 0。注意主对角线上的两个数交换位置，另外两个元素改变符号。
For A = [2 −1; 0 5], det(A) = 10, so A⁻¹ = (1/10) × [5 1; 0 2] = [0.5 0.1; 0 0.2]. Check: A × A⁻¹
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