📚 Matrices in IB and CCEA Mathematics: Key Points Review | IB CCEA 数学：矩阵 考点精讲
Matrices provide a powerful framework for organising data and performing operations that solve complex problems in mathematics, from systems of linear equations to geometric transformations. This revision guide brings together the essential concepts and techniques required for both IB Mathematics and CCEA Mathematics specifications, offering clear bilingual explanations with worked examples.
矩阵为整理数据、执行运算提供强大框架，解决从线性方程组到几何变换等复杂数学问题。本考点精讲汇集了 IB 数学和 CCEA 数学课程要求的基本概念与技巧，提供清晰的中英双解及实例。
1. Matrix Terminology and Notation | 矩阵术语与表示法
A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. We normally denote matrices by capital letters such as A, B, M. The order or dimension of a matrix is written as m × n, where m is the number of rows and n is the number of columns.
矩阵是由行和列排成的数字、符号或表达式的矩形阵列。通常用大写字母 A、B、M 表示。矩阵的阶或维度记作 m × n，其中 m 为行数，n 为列数。
Each entry in a matrix is called an element. For matrix A, the element in the ith row and jth column is denoted by aij. For example, a23 is the element in the second row and third column.
矩阵中的每个数称为元素。对于矩阵 A，位于第 i 行第 j 列的元素记作 aij。例如 a23 是第二行第三列的元素。
A general 2×2 matrix can be written as A = [a11 a12; a21 a22] or simply as [a b; c d]. The identity matrix I is a square matrix with ones on the main diagonal and zeros elsewhere, e.g. I2 = [1 0; 0 1]. A zero matrix O has all elements equal to zero.
一般的 2×2 矩阵可记作 A = [a11 a12; a21 a22] 或简写为 [a b; c d]。单位矩阵 I 是主对角线元素为 1、其余为 0 的方阵，如 I2 = [1 0; 0 1]。零矩阵 O 的所有元素均为零。
Two matrices are equal if they have the same order and all corresponding elements are equal. This means A = B only when aij = bij for every i and j.
若两矩阵同阶且对应元素全部相等，则两矩阵相等。即仅当对所有 i, j 都有 aij = bij 时，A = B。
2. Addition and Subtraction of Matrices | 矩阵加减法
Matrices can be added or subtracted only if they have the same order. The result is found by adding or subtracting the corresponding elements. For example, if A = [1 3; 2 −1] and B = [4 0; −2 5], then A + B = [1+4, 3+0; 2+(−2), −1+5] = [5 3; 0 4].
只有当矩阵同阶时才能进行加减法。结果由对应元素相加或相减得到。例如 A = [1 3; 2 −1] 与 B = [4 0; −2 5]，则 A + B = [1+4, 3+0; 2+(−2), −1+5] = [5 3; 0 4]。
Matrix addition is commutative: A + B = B + A, and associative: (A + B) + C = A + (B + C). The zero matrix acts as the additive identity: A + O = A.
矩阵加法满足交换律：A + B = B + A，结合律：(A + B) + C = A + (B + C)。零矩阵是加法单位元：A + O = A。
3. Scalar Multiplication | 标量乘法
Scalar multiplication involves multiplying every element of a matrix by a constant k. If A = [aij], then kA = [k × aij]. For instance, 3 × [1 −2; 0 4] = [3 −6; 0 12].
标量乘法指用常数 k 乘以矩阵的每个元素。若 A = [aij]，则 kA = [k × aij]。例如 3 × [1 −2; 0 4] = [3 −6; 0 12]。
The distributive laws hold: k(A + B) = kA + kB, and (k + m)A = kA + mA. Scalar multiplication is straightforward but crucial when working with inverses and transformations.
分配律成立：k(A + B) = kA + kB，且 (k + m)A = kA + mA。标量乘法虽简单，但在求逆和变换中至关重要。
4. Matrix Multiplication | 矩阵乘法
Matrix multiplication AB is defined only when the number of columns in A equals the number of rows in B. If A is m × n and B is n × p, the product AB is an m × p matrix. The element in row i, column j is given by cij = Σk=1n aik bkj.
矩阵乘法 AB 仅在 A 的列数等于 B 的行数时有定义。若 A 为 m × n，B 为 n × p，则乘积 AB 是 m × p 矩阵。第 i 行第 j 列元素为 cij = Σk=1n aik bkj。
Consider A = [2 1; 0 −1] and B = [3 0; 1 2]. The product AB = [2×3+1×1, 2×0+1×2; 0×3+(−1)×1, 0×0+(−1)×2] = [7 2; −1 −2]. Notice that the order matters: BA usually differs from AB; matrix multiplication is not commutative.
以 A = [2 1; 0 −1] 和 B = [3 0; 1 2] 为例。乘积 AB = [2×3+1×1, 2×0+1×2; 0×3+(−1)×1, 0×0+(−1)×2] = [7 2; −1 −2]。注意顺序很重要：BA 一般与 AB 不同，矩阵乘法不满足交换律。
Multiplication is associative: (AB)C = A(BC), and distributive: A(B + C) = AB + AC. The identity matrix I behaves like the number 1: AI = IA = A for any square matrix A of compatible size.
乘法满足结合律：(AB)C = A(BC)，分配律：A(B + C) = AB + AC。单位矩阵 I 类似于数字 1：对于任意同阶方阵 A，均有 AI = IA = A。
5. 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 defined as det(A) = ad − bc. It is a scalar value that indicates whether the matrix is singular (non-invertible) or non-singular (invertible).
det(A) = ad − bc
对于 2×2 矩阵 A = [a b; c d]，行列式定义为 det(A) = ad − bc。这一标量值可判断矩阵是否奇异（不可逆）或非奇异（可逆）。
If det(A) = 0, the matrix A is singular and has no inverse. If det(A) ≠ 0, the matrix is invertible. The determinant also appears in area scaling factors for geometric transformations.
若 det(A) = 0，则矩阵 A 奇异且无逆矩阵；若 det(A) ≠ 0，则矩阵可逆。行列式还在几何变换中表现为面积缩放因子。
Example: For M = [5 2; 3 4], det(M) = 5×4 − 2×3 = 20 − 6 = 14, so M is non-singular.
示例：M = [5 2; 3 4]，det(M) = 5×4 − 2×3 = 20 − 6 = 14，因此 M 为非奇异矩阵。
6. Inverse of a 2×2 Matrix | 2×2 矩阵的逆
If det(A) ≠ 0, the inverse matrix A−1 is given by swapping a and d, changing the signs of b and c, and dividing by the determinant:
A−1 = (1/(ad − bc)) [d −b; −c a]
若 det(A) ≠ 0，逆矩阵 A−1 由交换 a、d 位置，改变 b、c 符号并除以行列式得到：
A−1 = (1/(ad − bc)) [d −b; −c a]
The product A A−1 = I and A−1 A = I. The inverse is unique. For the matrix M above, M−1 = 1/14 [4 −2; −3 5] = [2/7 −1/7; −3/14 5/14].
乘积 A A−1 = I 且 A−1 A = I，逆矩阵唯一。对于上例 M，M−1 = 1/14 [4 −2; −3 5] = [2/7 −1/7; −3/14 5/14]。
Only square matrices can have inverses, and they must have a non-zero determinant. This concept is fundamental for solving systems of linear equations.
只有方阵才可能有逆矩阵，且必须行列式不为零。这一概念是解线性方程组的基础。
7. Solving Simultaneous Equations Using Matrices | 利用矩阵解方程组
A system of linear equations can be written in matrix form AX = B, where A is the coefficient matrix, X is the column vector of variables, and B is the constant vector. If A is invertible, the unique solution is X = A−1 B.
线性方程组可写成矩阵形式 AX = B，其中 A 为系数矩阵，X 为变量列向量，B 为常数项列向量。若 A 可逆，则唯一解为 X = A−1 B。
For example, solve 2x + y = 5 and −x + 3y = 1. Matrix form: [2 1; −1 3] [x; y] = [5; 1]. Compute det = (2)(3) − (1)(−1) = 6 + 1 = 7 ≠ 0. Inverse of A is 1/7 [3 −1; 1 2]. Then [x; y] = 1/7 [3 −1; 1 2] [5; 1] = 1/7 [15−1; 5+2] = [14/7; 7/7] = [2; 1]. So x = 2, y = 1.
例如求解 2x + y = 5 与 −x + 3y = 1。矩阵形式：[2 1; −1 3] [x; y] = [5; 1]。行列式 = (2)(3) − (1)(−1) = 7 ≠ 0。A−1 = 1/7 [3 −1; 1 2]。则 [x; y] = 1/7 [3 −1; 1 2] [5; 1] = 1/7 [14; 7] = [2; 1]。故 x = 2, y = 1。
This method extends to 3×3 systems, provided the coefficient matrix is invertible. In IB and CCEA exams, you may also be asked to use row reduction, but the inverse method is efficient for small systems.
该方法可推广至 3×3 方程组，前提是系数矩阵可逆。在 IB 和 CCEA 考试中，可能还要求使用行化简法，但对小型方程组，逆矩阵方法更为高效。
8. Matrix Transformations: Reflection, Rotation, Enlargement, Shear | 矩阵变换：反射、旋转、放大、剪切
In coordinate geometry, a 2×2 matrix M can represent a linear transformation of the plane. A point P with position vector [x; y] is mapped to P’ = M [x; y]. The following table summarises key transformation matrices under the mapping T(x,y) → (x’, y’).
在坐标几何中，2×2 矩阵 M 可表示平面的线性变换。位置向量为 [x; y] 的点 P 被映射到 P’ = M [x; y]。下表概括了 T(x,y) → (x’, y’) 下的主要变换矩阵。
| Transformation / 变换 | Matrix / 矩阵 |
|---|---|
| Identity / 恒等 | [1 0; 0 1] |
| Reflection in x-axis / 关于 x 轴反射 | [1 0; 0 −1] |
| Reflection in y-axis / 关于 y 轴反射 | [−1 0; 0 1] |
| Reflection in line y = x / 关于 y=x 反射 | [0 1; 1 0] |
| Reflection in line y = −x / 关于 y=−x 反射 | [0 −1; −1 0] |
| Rotation by θ (anticlockwise) / 逆时针旋转 θ | [cos θ −sin θ; sin θ cos θ] |
| Enlargement scale factor k / 放大 k 倍 | [k 0; 0 k] |
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