📚 GCSE CCEA Maths: Matrices Revision | GCSE CCEA 数学：矩阵 考点精讲
This article covers all the key concepts of matrices required for GCSE CCEA Mathematics, including operations, transformations, inverses, and solving equations. Each topic is explained with clear examples to aid revision.
本文涵盖了 GCSE CCEA 数学所需的所有矩阵关键概念，包括运算、变换、逆矩阵以及解方程。每个主题都配有清晰的示例以帮助复习。
1. What is a Matrix? | 什么是矩阵？
A matrix is a rectangular array of numbers (or symbols) arranged in rows and columns. Matrices are used to store and manipulate data, and to represent transformations in geometry. Each individual number inside a matrix is called an element.
矩阵是一个按行和列排列的数字（或符号）矩形阵列。矩阵用于存储和处理数据，并用于表示几何变换。矩阵内的每个单独数字称为元素。
For example, the matrix below has 2 rows and 3 columns:
例如，下面的矩阵有 2 行 3 列：
| 3 | -1 | 0 |
| 2 | 4 | 7 |
2. Order of a Matrix | 矩阵的阶
The order (or size) of a matrix is written as 'number of rows × number of columns' ng the order is essential before performing operations such as addition or multiplication.
矩阵的阶（或大小）写作“行数 × 列数”。一个具有 m 行 n 列的矩阵阶数为 m × n。在执行加法或乘法等运算之前，了解阶数至关重要。
ith the same number of rows and columns is called a square matrix, such as a 2 × 2 matrix.
例如，上面的矩阵阶数为 2 × 3。行数与列数相等的矩阵称为方阵，例如 2 × 2 矩阵。
3. Matrix Addition and Subtraction | 矩阵的加法和减法
Matrices can be added or subtracted only if they have the same order. The result is obtained by adding or subtracting corresponding elements. The order of the resulting matrix remains the same.
只有当矩阵阶数相同时，才可以进行加法或减法。结果通过对应元素相加或相减得到。结果矩阵的阶数保持不变。
For matrices A and B of the same order, (A + B)ij = Aij + Bij and (A – B)ij = Aij – Bij. The operation is not defined for matrices of different orders.
对于相同阶的矩阵 A 和 B，(A + B)ij = Aij + Bij，(A – B)ij = Aij – Bij。不同阶的矩阵之间未定义该运算。
Example: If
示例：如果
A =
| 1 | 3 |
| 2 | 4 |
| 5 | 1 |
| 0 | -2 |
| 6 | 4 |
| 2 | 2 |
则 A + B =
| 6 | 4 |
| 2 | 2 |
4. Scalar Multiplication | 标量乘法
Scalar multiplication involves multiplying a matrix by a single number (a scalar). Each element of the matrix is multiplied by the scalar. This operation works for matrices of any order.
标量乘法是将一个矩阵乘以一个单一数字（标量）。矩阵中的每个元素都与该标量相乘。该运算适用于任何阶的矩阵。
If k is a scalar and A is a matrix, the resulting matrix kA has elements k × Aij. The order remains unchanged.
如果 k 是标量，A 是矩阵，结果矩阵 kA 的元素为 k × Aij。阶数保持不变。
Example: For A =
| 2 | -1 |
| 0 | 3 |
| 6 | -3 |
| 0 | 9 |
示例：对于 A =
| 2 | -1 |
| 0 | 3 |
| 6 | -3 |
| 0 | 9 |
5. Matrix Multiplication | 矩阵乘法
ows in the second matrix. If A is m × n and B is n × p, the product AB is an m × p matrix.
矩阵乘法仅在第一个矩阵的列数等于第二个矩阵的行数时定义。如果 A 是 m × n，B 是 n × p，则乘积 AB 是 m × p 矩阵。
Each element in the product is found by multiplying the elements of a row of A with the corresponding elements of a column of B, then adding the products. This is the dot product.
乘积中的每个元素是通过将 A 某行的元素与 B 某列的对应元素相乘，再将乘积累加得到的。这就是点积。
Example: Multiply A =
| 1 | 2 |
| 3 | 4 |
| 5 | 6 |
| 7 | 8 |
示例：计算 A =
| 1 | 2 |
| 3 | 4 |
| 5 | 6 |
| 7 | 8 |
AB =
| (1×5+2×7) | (1×6+2×8) |
| (3×5+4×7) | (3×6+4×8) |
| 19 | 22 |
| 43 | 50 |
Note that matrix multiplication is not commutative: AB ≠ BA in general.
注意矩阵乘法不满足交换律：一般来说 AB ≠ BA。
6. Multiplying a Matrix by a Vector | 矩阵与向量的乘法
lumn vector gives a 2 × 1 column vector. This operation is fundamental in transformations.
列向量可视为只有一列的矩阵。2 × 2 矩阵乘以 2 × 1 列向量得到一个 2 × 1 列向量。该运算在变换中十分基础。
If M =
| a | b |
| c | d |
| x |
| y |
| ax + by |
| cx + dy |
如果 M =
| a | b |
| c | d |
| x |
| y |
| ax + by |
| cx + dy |
7. Transformations in the Plane | 平面内的变换
Matrices can represent geometric transformations such as rotations, reflections, enlargements, and combinations of these. A point (x, y) is written as a column vector, and the image is found by multiplying the transformation matrix by that column vector.
矩阵可以表示几何变换，例如旋转、反射、放大以及这些的组合。点 (x, y) 写作列向量，像点通过将变换矩阵乘以该列向量求得。
The transformation is applied to every point of a shape. To find the image of a shape, apply the matrix to each vertex's position vector, or use the matrix to map the general point (x, y).
变换应用于形状的每个点。要找到形状的像，只需
Published by TutorHao | GCSE Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图