📚 Matrices: Operations, Determinants, Inverses and Linear Transformations | 矩阵：运算、行列式、逆与线性变换
Matrices are a core tool in AQA A-Level Further Mathematics. They are used to solve simultaneous linear equations, to describe geometric transformations of the plane, and to model structures in economics, computer graphics and engineering. This guide covers the essential definitions, the arithmetic of matrices, determinants, inverses, linear transformations, and the exam techniques that turn these ideas into marks.
矩阵是 AQA A-Level 进阶数学的核心工具，用于求解线性方程组、描述平面上的几何变换，以及在经济学、计算机图形学和工程学中为结构建模。本指南涵盖基本定义、矩阵运算、行列式、逆矩阵、线性变换，以及把这些概念转化为分数的考试技巧。
1. Order and Notation | 阶数与记号
A matrix is a rectangular array of numbers arranged in rows and columns. In AQA A-Level Further Mathematics, the order (sometimes called the size) of a matrix is written as m × n umber of columns. The order is always quoted as rows × columns, never the other way round.
矩阵是由数字按行和列排列成的矩形阵列。在 AQA A-Level 进阶数学中，矩阵的阶数（有时也称大小）写作 m × n，其中 m 是行数，n 是列数。阶数一律写成"行 × 列"，绝不能颠倒顺序。
An individual entry is labelled with a lowercase letter and two subscripts: aᵢⱼ is the entry in row i and column j. The first subscript always gives the row, the second always gives the column.
单个元素用小写字母加两个下标表示：aᵢⱼ 表示第 i 行、第 j 列的元素。第一个下标永远是行，第二个下标永远是列。
For example, A = [ 2 −1 ; 3 4 ] is a 2 × 2 matrix. The entry in row 1, column 2 is a₁₂ = −1, and the entry in row 2, column 1 is a₂₁ = 3. The entry a₂₂ = 4 and a₁₁ = 2.
例如，A = [ 2 −1 ; 3 4 ] 是一个 2 × 2 阶矩阵。第 1 行第 2 列的元素写作 a₁₂ = −1，第 2 行第 1 列的元素是 a₂₁ = 3，另外 a₂₂ = 4，a₁₁ = 2。
- A matrix with the same number of rows and columns is called a square matrix. | 行数与列数相同的矩阵称为方阵。
- A 1 × n matrix is a row vector; an m × 1 matrix is a column vector. | 1 × n 矩阵称为行向量；m × 1 矩阵称为列向量。
- Matrices are usually written with capital letters (A, B, M) and their entries with the matching lowercase letter. | 矩阵通常用大写字母（A、B、M）表示，其元素用对应的小写字母表示。
2. Addition, Subtraction and Scalar Multiplication | 加法、减法与标量乘法
Two matrices can be added or subtracted only if they have exactly the same order. Addition is performed entry by entry: the entry in position (i, j) of A + B is aᵢⱼ + bᵢⱼ. Subtraction works in exactly the same way, using aᵢⱼ − bᵢⱼ.
两个矩阵只有在阶数完全相同时才能相加或相减。加法是逐元素进行的：A + B 在第 (i, j) 位置的元素就是 aᵢⱼ + bᵢⱼ。减法同理，用 aᵢⱼ − bᵢⱼ 计算。
Multiplying a matrix by a scalar (an ordinary number) means multiplying every entry by that scalar. If A = [ 2 −1 ; 3 4 ], then 3A = [ 6 −3 ; 9 12 ]. A common exam slip is to multiply only the first row or only the diagonal entries.
矩阵乘以标量（普通数）意味着把每一个元素都乘以该数。若 A = [ 2 −1 ; 3 4 ]，则 3A = [ 6 −3 ; 9 12 ]。常见错误是只乘第一行，或只乘对角线上的元素。
With B = [ 1 0 ; 2 5 ] we get A + B = [ 3 −1 ; 5 9 ] and A − B = [ 1 −1 ; 1 −1 ] ) = (A + B) + C), and scalar multiplication distributes over addition: k(A + B) = kA + kB.
取 B = [ 1 0 ; 2 5 ]，则 A + B = [ 3 −1 ; 5 9 ]，A − B = [ 1 −1 ; 1 −1 ]。矩阵加法满足交换律（A + B = B + A）与结合律（A + (B + C) = (A + B) + C），标量乘法对加法满足分配律：k(A + B) = kA + kB。
3. Matrix Multiplication | 矩阵乘法
Matrix multiplication is defined only when the number of columns of the first matrix equals the number of rows of the second matrix has order m × p. If the inner dimensions do not match, the product simply does not exist.
矩阵乘法只有在第一个矩阵的列数等于第二个矩阵的行数时才有定义。若 A 的阶数为 m × n，B 的阶数为 n × p，则 AB 的阶数为 m × p。若内侧维度不匹配，乘积就不存在。
The entry in row i and column j of AB is found by multiplying the entries of row i of A with the matching entries of column j of B and adding the results:
AB 中第 i 行第 j 列的元素，是把 A 的第 i 行与 B 的第 j 列对应元素相乘后再相加得到的：
(AB)ᵢⱼ = Σₖ aᵢₖ bₖⱼ
Using A = [ 2 −1 ; 3 4 ] and B = [ 1 0 ; 2 5 ]: the top-left entry of AB is 2 × 1 + (−1) × 2 = 0, so
以 A = [ 2 −1 ; 3 4 ] 和 B = [ 1 0 ; 2 5 ] 为例：AB 左上角元素为 2 × 1 + (−1) × 2 = 0，于是
AB = [ 0 −5 ; 11 20 ] BA = [ 2 −1 ; 19 18 ]
Notice that AB ≠ BA, so matrix multiplication is not commutative. Order matters both in the arithmetic and in the geometry of transformations.
注意 AB ≠ BA，因此矩阵乘法不满足交换律。顺序在运算中与在几何变换中同样重要。
- AB = O does not imply A = O or B = O. | AB = O 不能推出 A = O 或 B = O。
- Matrix multiplication is associative: (AB)C = A(BC). | 矩阵乘法满足结合律：(AB)C = A(BC)。
- It distributes over addition: A(B + C) = AB + AC. | 它对加法满足分配律：A(B + C) = AB + AC。
4. Identity, Zero and Transpose | 单位矩阵、零矩阵与转置
The identity matrix I is the square matrix with 1s on the leading diagonal and 0s elsewhere. For 2 × 2 we have I = [ 1 0 ; 0 1 ] mber 1 in ordinary arithmetic: AI = IA = A for every matrix A that can be multiplied by I.
单位矩阵 I 是主对角线上全为 1、其余位置全为 0 的方阵。2 × 2 时 I = [ 1 0 ; 0 1 ]。它的作用类似普通算术中的数字 1：对所有能与 I 相乘的矩阵 A，都有 AI = IA = A。
The zero matrix t is the additive identity, so A + O = A. Note that there is a zero matrix for each order.
零矩阵 O 的所有元素都是 0，它是加法的单位元，因此 A + O = A。注意每种阶数都有自己的零矩阵。
The transpose Aᵀ s obtained by turning rows into columns. Transposing twice returns the original matrix, (Aᵀ)ᵀ = A. A useful algebraic rule is (AB)ᵀ = BᵀAᵀ - the order reverses. A square matrix satisfying A = Aᵀ is called symmetric, for example [ 1 3 ; 3 7 ].
转置 Aᵀ 是把行变成列得到的矩阵。转置两次回到原矩阵，(Aᵀ)ᵀ = A。一个常用的代数规则是 (AB)ᵀ = BᵀAᵀ——顺序要反过来。满足 A = Aᵀ 的方阵称为对称矩阵，例如 [ 1 3 ; 3 7 ]。
5. Determinants of 2 × 2 and 3 × 3 Matrices | 2 × 2 与 3 × 3 矩阵的行列式
gle number associated with a square matrix. For a 2 × 2 matrix with entries in the pattern [ a b ; c d ], the determinant is the difference of the two diagonal products:
行列式是与方阵对应的一个数。对于按 [ a b ; c d ] 排列的 2 × 2 矩阵，行列式等于两条对角线乘积之差：
det A = ad − bc
For A = [ 2 −1 ; 3 4 ]: det A = 2 × 4 − (−1) × 3 = 8 + 3 = 11. Remember to subtract the product bc, not to add it - sign errors here cost the whole question.
对 A = [ 2 −1 ; 3 4 ]：det A = 2 × 4 − (−1) × 3 = 8 + 3 = 11。记住是减去 bc 的乘积，而不是加上它——这里符号出错会导致整题失分。
× 3 matrix, expand along the first row using 2 × 2 minors with alternating signs +, −, +:
对于 3 × 3 矩阵，沿第一行按 +、−、+ 交替符号展开为 2 × 2 余子式：
det = a₁₁(a₂₂a₃₃ − a₂₃a₃₂) − a₁₂(a₂₁a₃₃ − a₂₃a₃₁) + a₁₃(a₂₁a₃₂ − a₂₂a₃₁)
Applying this to M = [ 1 2 3 ; 0 1 4 ; 5 6 0 ] gives 1(−24) − 2(−20) + 3(−5) = −24 + 40 − 15 = 1.
把它用于 M = [ 1 2 3 ; 0 1 4 ; 5 6 0 ]，得到 1(−24) − 2(−20) + 3(−5) = −24 + 40 − 15 = 1。
- det (AB) = det A × det B. | det (AB) = det A × det B。
- det (Aᵀ) = det A. | det (Aᵀ) = det A。
- For an n × n matrix, det (kA) = kⁿ det A. | 对 n × n 矩阵，det (kA) = kⁿ det A。
6. The Inverse of a 2 × 2 Matrix | 2 × 2 矩阵的逆
The inverse A⁻¹ of a square matrix A satisfies A A⁻¹ = A⁻¹ A = I. Only matrices with a non-zero determinant have an inverse. For a 2 × 2 matrix [ a b ; c d ] the formula is short enough to memorise exactly:
方阵 A 的
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