📚 Matrices | 矩阵
Matrices are rectangular arrays of numbers that encode linear systems, transformations and more. In AQA A-Level Further Mathematics, matrix algebra appears across Core Pure: you must be confident with order, multiplication, determinants, inverses, simultaneous equations and geometric transformations.
矩阵是由数字或代数式排列成的矩形数表，用来表示线性方程组、几何变换等。在 AQA A-Level 进阶数学中，矩阵代数贯穿核心纯数部分，你需要熟练掌握矩阵阶数、乘法、行列式、逆矩阵、联立方程组与几何变换。
1. Matrix Notation and Order | 矩阵记法与阶数
A general 2 × 2 matrix is written as A = [a b; c d], where a, b, c and d are the elements.
矩阵是由数字（称为元素）按行和列排列成的矩形数表。矩阵的阶数写成“行 × 列”，因此 3 × 2 矩阵有 3 行、2 列。一般的 2 × 2 矩阵写作 A = [a b; c d]，其中 a、b、c、d 是矩阵元素。
To refer to a specific element, use a subscript position: aᵢⱼ is the element in row i and column j. Two matrices are equal only when they have the same order and all corresponding elements are equal.
若要表示某个特定元素，可用下标表示位置：aᵢⱼ 是第 i 行第 j 列的元素。两个矩阵只有阶数相同且所有对应元素都相等时才相等。
A = [a b; c d] is a 2 × 2 matrix | A = [a b; c d] 是一个 2 × 2 矩阵
2. Addition, Subtraction and Scalar Multiplication | 矩阵加法、减法与数乘
You can only add or subtract matrices when they have exactly the same order. Add or subtract corresponding elements. The result has the same order as the original matrices.
只有阶数完全相同的矩阵才能相加或相减。加法与减法都是对应元素相加或相减，所得矩阵的阶数与原矩阵相同。
If A = [a b; c d] and B = [p q; r s], then A + B = [a + p b + q; c + r d + s]
若 A = [a b; c d]，B = [p q; r s]，则 A + B = [a + p b + q; c + r d + s]
Scalar multiplication means multiplying every element of a matrix by a number k. For example, kA is formed by replacing each entry aᵢⱼ with k aᵢⱼ. Matrix subtraction is just addition with the scalar multiple −1.
数乘是指用数 k 乘以矩阵中的每一个元素。例如 kA 就是把每个元素 aᵢⱼ 替换为 k aᵢⱼ。矩阵减法等价于加上 −1 倍的数乘结果。
3. Matrix Multiplication | 矩阵乘法
multiplication is not commutative in general: AB and BA may be different, and one product may not even be defined.
矩阵乘积 AB 只有在 A 的列数等于 B 的行数时才有定义。若 A 是 m × n 矩阵，B 是 n × p 矩阵，则 AB 是 m × p 矩阵。矩阵乘法一般不满足交换律：AB 与 BA 可能不同，甚至其中一个乘积可能不存在。
A = [a b; c d], B = [p q; r s] ⇒ AB = [ap + br aq + bs; cp + dr cq + ds]
A = [a b; c d]，B = [p q; r s] ⇒ AB = [ap + br aq + bs; cp + dr cq + ds]
7 8], then AB = [19 22; 43 50]. Notice that BA = [23 34; 31 46], so AB ≠ BA in this case.
例如，若 A = [1 2; 3 4]，B = [5 6; 7 8]，则 AB = [19 22; 43 50]。注意 BA = [23 34; 31 46]，因此这里 AB ≠ BA。
4. Zero and Identity Matrices | 零矩阵与单位矩阵
: A + O = A. The identity matrix, written I, has 1s on the main diagonal and 0s elsewhere.
零矩阵记作 O，所有元素都为 0。任何同阶矩阵 A 加上零矩阵都不变：A + O = A。单位矩阵记作 I，主对角线上的元素为 1，其余元素为 0。
I₂ = [1 0; 0 1]
I₂ = [1 0; 0 1]
the products are defined. For any 2 × 2 matrix A, multiplication by I₂ leaves A unchanged.
单位矩阵在矩阵代数中的作用类似于数字 1：只要乘积有定义，就有 AI = IA = A。对任意 2 × 2 矩阵 A，左乘或右乘 I₂ 都不会改变 A。
5. Determinant of a 2 × 2 Matrix | 2 × 2 矩阵的行列式
− bc. The determinant is a scalar value that tells you whether the matrix has an inverse.
对于 2 × 2 矩阵 M = [a b; c d]，行列式为 det M = ad − bc。行列式是一个标量值，用来判断矩阵是否有逆矩阵。
det M = ad − bc
det M = ad − bc
ists. For example, M = [2 5; 1 3] has determinant 2 × 3 − 5 × 1 = 1, so M is non-singular.
若 det M = 0，矩阵为奇异矩阵，没有逆矩阵。若 det M ≠ 0，矩阵为非奇异矩阵，存在逆矩阵。例如 M = [2 5; 1 3] 的行列式为 2 × 3 − 5 × 1 = 1，因此 M 是非奇异的。
6. Inverse of a 2 × 2 Matrix | 2 × 2 矩阵的逆矩阵
ly when det M ≠ 0. The inverse is found by swapping a and d, changing the signs of b and c, and dividing by the determinant.
2 × 2 矩阵 M = [a b; c d] 只有在 det M ≠ 0 时才有逆矩阵。求逆时交换 a 和 d，改变 b 和 c 的符号，再除以行列式。
M⁻¹ = 1/(ad − bc) [d −b; −c a]
M⁻¹ = 1/(ad − bc) [d −b; −c a]
I. If det M = 0, M is singular and no inverse exists, so do not attempt to divide by zero.
求出逆矩阵后应验证 M M
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