📚 A-Level Mathematics: Matrices Intensive Revision | A-Level 数学：矩阵 考点精讲
Matrices are a fundamental tool in A-Level Mathematics, providing a compact way to organise numbers and perform operations such as transformations and solving simultaneous equations. This revision guide covers all key topics from matrix notation to determinants, inverses, and applications, equipping you with the skills needed for exam success.
矩阵是 A-Level 数学中的基础工具，能简洁地组织数据并进行变换、解联立方程等运算。本考点精讲涵盖从矩阵记法到行列式、逆矩阵及应用的所有核心内容，助你掌握考试所需的技能。
1. Defining Matrices and Notation | 矩阵的定义与记法
A matrix is a rectangular array of numbers or expressions arranged in rows and columns. The order of a matrix is given by m × n, where m is the number of rows and n is the number of columns. Elements are typically denoted as aij, where i is the row index and j is the column index. For example, matrix A = [1 2 3; 4 5 6] has order 2 × 3.
矩阵是一个按行和列排列的矩形数字或表达式阵列。矩阵的阶由 m × n 给出，其中 m 是行数，n 是列数。元素通常记作 aij，i 为行索引，j 为列索引。例如矩阵 A = [1 2 3; 4 5 6] 是 2 × 3 阶。
Two matrices are equal if they have the same order and all corresponding elements are equal. A row matrix has only one row, while a column matrix has only one column. A square matrix has the same number of rows and columns.
两个矩阵若阶数相同且所有对应元素相等，则它们相等。行矩阵只有一行，列矩阵只有一列。方阵的行数和列数相等。
2. Addition and Subtraction of Matrices | 矩阵的加法与减法
Matrices of the same order can be added or subtracted by combining corresponding elements. For matrices A and B both of order m × n, (A ± B)ij = aij ± bij. If A = [1 2; 3 4] and B = [5 6; 7 8], then A + B = [6 8; 10 12].
同阶矩阵可通过对应元素相加减来完成加法或减法。对于均为 m × n 阶的矩阵 A 和 B，有 (A ± B)ij = aij ± bij。若 A = [1 2; 3 4]，B = [5 6; 7 8]，则 A + B = [6 8; 10 12]。
Matrix addition is commutative and associative: A + B = B + A and (A + B) + C = A + (B + C). Subtraction is simply the addition of the negative: A – B = A + (-B).
矩阵加法满足交换律和结合律：A + B = B + A，(A + B) + C = A + (B + C)。减法可看作加上负矩阵：A – B = A + (-B)。
3. Scalar Multiplication | 标量乘法
Multiplying a matrix by a scalar k means multiplying every element by k: kA = [k aij]. For A = [2 -1; 0 3], 3A = [6 -3; 0 9]. Scalar multiplication distributes over matrix addition: k(A + B) = kA + kB.
矩阵乘以标量 k 意味着将每个元素乘以 k：kA = [k aij]。对于 A = [2 -1; 0 3]，3A = [6 -3; 0 9]。标量乘法对矩阵加法满足分配律：k(A + B) = kA + kB。
You can also combine scalar multiplication with matrix multiplication, respecting the associative property: k(AB) = (kA)B = A(kB).
标量乘法可与矩阵乘法结合，满足结合律：k(AB) = (kA)B = A(kB)。
4. Matrix Multiplication | 矩阵乘法
Two matrices A (m × n) and B (n × p) can be multiplied to give C = AB of order m × p. The element cij is the dot product of the i-th row of A and the j-th column of B: cij = Σk aik bkj. For example, A = [1 2; 3 4], B = [2 0; 1 2]: AB = [1×2+2×1, 1×0+2×2; 3×2+4×1, 3×0+4×2] = [4 4; 10 8].
矩阵 A (m × n) 与 B (n × p) 可相乘得到 m × p 阶矩阵 C = AB。元素 cij 是 A 的第 i 行与 B 的第 j 列的点积：cij = Σk aik bkj。例如 A = [1 2; 3 4]，B = [2 0; 1 2]：AB = [1×2+2×1, 1×0+2×2; 3×2+4×1, 3×0+4×2] = [4 4; 10 8]。
Matrix multiplication is not commutative; in general AB ≠ BA. It is associative: (AB)C = A(BC), and distributive over addition: A(B + C) = AB + AC. For a square matrix A, powers are defined as A² = AA, A³ = A²A, etc.
矩阵乘法不满足交换律，通常 AB ≠ BA。但满足结合律：(AB)C = A(BC)，且对加法有分配律：A(B + C) = AB + AC。对于方阵 A，可定义幂次：A² = AA，A³ = A²A，以此类推。
5. Identity and Zero Matrices | 单位矩阵与零矩阵
The n × n identity matrix In has 1s on the main diagonal and 0s elsewhere. It satisfies AI = IA = A for any conformable matrix A. The zero matrix O has all elements zero; A + O = A and AO = O (where defined). For example, I₂ = [1 0; 0 1], I₃ = [1 0 0; 0 1 0; 0 0 1].
n 阶单位矩阵 In 主对角线元素为 1，其余为 0。对相容的矩阵 A，有 AI = IA = A。零矩阵 O 元素全为 0；满足 A + O = A，且若乘法有定义则 AO = O。例如 I₂ = [1 0; 0 1]，I₃ = [1 0 0; 0 1 0; 0 0 1]。
The identity matrix plays a role analogous to the number 1 in ordinary multiplication, while the zero matrix behaves like 0.
单位矩阵在矩阵乘法中的作用类似于数字 1，零矩阵则类似于 0。
6. Determinant of a 2×2 Matrix | 2×2 矩阵的行列式
For a 2×2 matrix A = [a b; c d], the determinant is det(A) = |A| = ad – bc. The determinant is a scalar that indicates whether the matrix is invertible: a non-zero determinant means A⁻¹ exists. If A = [3 4; 2 1], det(A) = 3×1 – 4×2 = 3 – 8 = -5.
对于 2×2 矩阵 A = [a b; c d]，行列式为 det(A) = |A| = ad – bc。行列式是一个标量，指示矩阵是否可逆：非零时 A⁻¹ 存在。若 A = [3 4; 2 1]，则 det(A) = 3×1 – 4×2 = 3 – 8 = -5。
A matrix with zero determinant is called singular and does not have an inverse. The determinant of a product satisfies det(AB) = det(A) det(B).
行列式为零的矩阵称为奇异矩阵，没有逆。乘积的行列式满足 det(AB) = det(A) det(B)。
7. Inverse of a 2×2 Matrix | 2×2 矩阵的逆
If det(A) ≠ 0 for A = [a b; c d], its inverse is A⁻¹ = (1/det(A)) [d -b; -c a]. To verify: AA⁻¹ = I. For example, A = [2 3; 1 4]: det =
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