IB & WJEC Mathematics: Matrices Revision Guide | IB与WJEC数学：矩阵考点精讲

📚 IB & WJEC Mathematics: Matrices Revision Guide | IB与WJEC数学：矩阵考点精讲

Matrices are a cornerstone of linear algebra and appear in both IB Mathematics (particularly the Applications and Interpretation HL course) and WJEC Further Mathematics specifications. This revision guide brings together the essential theory, key operations, and typical exam-style applications to help you tackle matrix problems with confidence.

矩阵是线性代数的基石，在IB数学（尤其是应用与解释HL课程）和WJEC进阶数学考试中均有重要地位。本考点精讲汇集了核心理论、关键运算以及典型的考试题型，助你从容应对矩阵相关问题。

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1. Introduction to Matrices | 矩阵入门

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. The size of a matrix is called its order and is written as m × n, where m is the number of rows and n is the number of columns. An element in row i and column j is often denoted as a_ij.

矩阵是由数字、符号或表达式按行和列排列成的矩形阵列。矩阵的大小称为阶数，记作 m × n，其中 m 是行数，n 是列数。位于第 i 行第 j 列的元素通常记作 a_ij。

Example: The matrix B = [5 -1 0; 2 4 7] has order 2 × 3. The element b₂₃ (row 2, column 3) is 7.

示例：矩阵 B = [5 -1 0; 2 4 7] 为 2 × 3 阶。元素 b₂₃（第 2 行第 3 列）是 7。

• Square matrix: a matrix with the same number of rows and columns (e.g., 2×2, 3×3).

• 方阵：行数与列数相等的矩阵（如 2×2、3×3）。

• Zero matrix: every entry is 0; it is usually denoted by O.

• 零矩阵：所有元素均为 0；通常记作 O。

• Identity matrix: a square matrix with 1s on the main diagonal and 0s elsewhere. The 2×2 identity matrix is I = [1 0; 0 1].

• 单位矩阵：主对角线全为 1、其余元素为 0 的方阵。2×2 单位矩阵为 I = [1 0; 0 1]。

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2. Matrix Operations | 矩阵运算

Matrices of the same order can be added or subtracted by combining the corresponding entries. Scalar multiplication involves multiplying every element of the matrix by the scalar constant.

同阶矩阵可以通过对应元素相加或相减来实现加减法运算。标量乘法则是将矩阵的每一个元素都乘以该标量常数。

Addition and subtraction: If A = [a_ij] and B = [b_ij], then A ± B = [a_ij ± b_ij].

加减法：若 A = [a_ij] 且 B = [b_ij]，则 A ± B = [a_ij ± b_ij]。

Example: Let A = [2 3; 1 -4] and B = [0 5; -2 1]. Then A + B = [2+0 3+5; 1+(-2) -4+1] = [2 8; -1 -3].

示例：设 A = [2 3; 1 -4]，B = [0 5; -2 1]，则 A + B = [2+0 3+5; 1+(-2) -4+1] = [2 8; -1 -3]。

Scalar multiplication: kA = [k a_ij]. For instance, 3A = [6 9; 3 -12].

标量乘法：kA = [k a_ij]。例如 3A = [6 9; 3 -12]。

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3. Matrix Multiplication | 矩阵乘法

Two matrices can be multiplied only if the number of columns in the first matrix equals the number of rows in the second. If A is m × n and B is n × p, then the product AB is an m × p matrix. The element in row i, column j of AB is obtained by multiplying the elements of the i-th row of A with the corresponding elements of the j-th column of B and summing the results:

两个矩阵相乘的前提是左边矩阵的列数等于右边矩阵的行数。若 A 为 m × n 矩阵，B 为 n × p 矩阵，则乘积 AB 是 m × p 矩阵。AB 中第 i 行第 j 列的元素由 A 的第 i 行各元素与 B 的第 j 列对应元素相乘再求和得到：

(AB)_ij = ∑_k=1ⁿ a_ik b_kj

Example: A = [2 1; 0 3] (2×2), B = [1 4; 5 2] (2×2). Then AB = [2×1+1×5 2×4+1×2; 0×1+3×5 0×4+3×2] = [7 10; 15 6].

示例：A = [2 1; 0 3]，B = [1 4; 5 2]，则 AB = [2×1+1×5 2×4+1×2; 0×1+3×5 0×4+3×2] = [7 10; 15 6]。

Critical note: Matrix multiplication is not commutative; in general, AB ≠ BA. Always check the order of multiplication, especially in transformation problems.

重要提醒：矩阵乘法不满足交换律；通常情况下 AB ≠ BA。务必注意乘法的顺序，尤其是在变换问题中。

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4. Determinant and Inverse of a 2×2 Matrix | 2×2 矩阵的行列式与逆矩阵

For a 2×2 matrix A = [a b; c d], the determinant is defined as det(A) = ad − bc. The determinant determines whether the matrix is invertible (non-singular) or not (singular).

对于 2×2 矩阵 A = [a b; c d]，其行列式定义为 det(A) = ad − bc。行列式决定了矩阵是否可逆（非奇异）或不可逆（奇异）。

If det(A) ≠ 0, the inverse matrix is given by:

A⁻¹ = (1/det(A)) [d −b; −c a]

若 det(A) ≠ 0，逆矩阵由下式给出：

A⁻¹ = (1/det(A)) [d −b; −c a]

Example: Find the inverse of M = [4 7; 2 6]. det(M) = 4×6 − 7×2 = 24 − 14 = 10. Hence M⁻¹ = (1/10) [6 −7; −2 4] = [0.6 −0.7; −0.2 0.4].

示例：求 M = [4 7; 2 6] 的逆矩阵。det(M) = 4×6 − 7×2 = 10。因此 M⁻¹ = (1/10) [6 −7; −2 4] = [0.6 −0.7; −0.2 0.4]。

If det(A) = 0, the matrix is singular and has no inverse. This also means the rows (or columns) are linearly dependent.

若 det(A) = 0，矩阵为奇异矩阵，没有逆矩阵。这也意味着矩阵的行（或列）线性相关。

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5. Solving Linear Systems Using Inverse Matrices | 利用逆矩阵求解线性方程组

A system of linear equations can be written in matrix form as AX = B, where A is the coefficient matrix, X is the column vector of variables, and B is the constant column vector. If A is non-singular, the unique solution is X = A⁻¹B.

线性方程组可以写成矩阵方程 AX = B，其中 A 是系数矩阵，X 是变量列向量，B 是常数项列向量。若 A

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