AS Mathematics: Matrices Exam Focus | AS 数学：矩阵考点精讲

📚 AS Mathematics: Matrices Exam Focus | AS 数学：矩阵考点精讲

Matrices are a cornerstone of AS Mathematics, essential for working with linear transformations, vectors, and systems of linear equations. This focused revision guide breaks down every key concept—from basic definitions to solving simultaneous equations using inverse matrices—and provides exam-winning strategies to help you avoid common mistakes.

矩阵是AS数学的基石，对于处理线性变换、向量和线性方程组至关重要。这份精讲复习指南分解了从基本定义到用逆矩阵解联立方程的所有核心概念，并提供制胜策略，帮助你避开常见错误。

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1. Matrix Definition & Order | 矩阵的定义与阶数

ngular array of numbers arranged in rows and columns. The dimension (or order) is given as 'number of rows × number of columns'. For instance, a 3 × 2 matrix has 3 rows and 2 columns.

矩阵是一个按行和列排列的矩形数字阵列。维数（或称阶数）表示为“行数×列数”。例如，一个3 × 2矩阵有3行2列。

Entries are denoted by a small letter with two subscripts: aᵢⱼ ces are usually named with capital letters A, B, C. A general 2×2 matrix can be written as

a₁₁	a₁₂
a₂₁	a₂₂

or more compactly

a	b
c	d

.

元素用带有两个下标的字母表示：aᵢⱼ是第i行第j列的元素。矩阵通常用大写字母A、B、C命名。一个一般的2×2矩阵可以写作

a₁₁	a₁₂
a₂₁	a₂₂

或更简洁地

a	b
c	d

.

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2. Addition & Subtraction | 矩阵的加法与减法

ey have the same order. Simply add or subtract corresponding elements. For example, if A =

a	b
c	d

and B =

e	f
g	h

, then A + B =

a+e	b+f
c+g	d+h

.

只有当两个矩阵阶数相同时，才能相加或相减。只需对应元素相加或相减。例如，若A =

a	b
c	d

且 B =

e	f
g	h

, 则 A + B =

a+e	b+f
c+g	d+h

.

s elements a−e, b−f, c−g, d−h. Matrix addition is commutative (A+B = B+A) and associative.

减法同理：A − B 得到元素 a−e, b−f, c−g, d−h。矩阵加法满足交换律（A+B = B+A）和结合律。

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3. Scalar Multiplication | 矩阵的数乘

tiply a matrix by a scalar (a constant k), multiply every entry of the matrix by k. If A =

a	b
c	d

, then kA =

ka	kb
kc	kd

.

一个矩阵与一个标量（常数k）相乘，就是将矩阵的每一项都乘以k。若 A =

a	b
c	d

, 则 kA =

ka	kb
kc	kd

.

atrix addition and vice versa. Negative of a matrix is obtained by multiplying by −1: −A =

−a	−b
−c	−d

.

数乘对矩阵加法具有分配律，反之亦然。一个矩阵的负矩阵可通过乘以−1得到：−A =

−a	−b
−c	−d

.

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4. Matrix Multiplication | 矩阵的乘法

p, the product AB is m × p. The entry in row i, column j of AB is found by multiplying corresponding elements from row i of A and column j of B and summing the results.

矩阵乘法仅在左矩阵的列数等于右矩阵的行数时才定义。若A是 m × n 矩阵，B是 n × p 矩阵，则积 AB 是 m × p 矩阵。AB中第i行第j列的元素是通过将A第i行与B第j列的对应元素相乘并求和得到的。

For 2×2 matrices A =

a	b
c	d

and B =

e	f
g	h

, AB =

ae+bg	af+bh
ce+dg	cf+dh

. Note that the top-left element comes from row1 · col1: a×e + b×g.

对于2×2矩阵 A =

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