📚 Matrices: Core Operations, Determinants and Transformations | 矩阵：核心运算、行列式与变换
metric transformations. This guide covers all the core matrix skills you need for the exam, with a focus on 2 × 2 matrices.
矩阵是 AQA A-Level 数学中最有力的工具之一。它们让你能够以紧凑的代数形式存储数据、求解联立方程组以及建立几何变换模型。本指南涵盖考试所需的所有核心矩阵技能，重点放在 2 × 2 矩阵上。
1. What Is a Matrix? | 什么是矩阵？
nd n is the number of columns. For example, A = [2 −1; 0 3] is a 2 × 2 matrix because it has two rows and two columns.
矩阵是按行和列排列的数字矩形阵列。矩阵的阶数写作 m × n，其中 m 是行数，n 是列数。例如，A = [2 −1; 0 3] 是一个 2 × 2 矩阵，因为它有两行和两列。
Each number inside a matrix is called an element or entry. We usually label elements by giving the row number first and the column number second. In the matrix above, the element in row 1, column 1 is 2, and the element in row 2, column 1 is 0.
矩阵中的每个数字称为元素或项。我们通常先给出行号，再给出列号来标记元素。在上面的矩阵中，第 1 行第 1 列的元素是 2，第 2 行第 1 列的元素是 0。
ry, b is the top-right entry, c is the bottom-left entry, and d is the bottom-right entry. This notation is used throughout the determinant and inverse formulae.
我们通常把一般的 2 × 2 矩阵写成 A = [a b; c d]。这里 a 是左上元素，b 是右上元素，c 是左下元素，d 是右下元素。这个记号会贯穿行列式与逆矩阵公式。
2. Matrix Addition and Subtraction | 矩阵加减法
. For example, if A = [1 4; −2 0] and B = [3 −1; 5 2], then A + B = [1+3 4+(−1); −2+5 0+2] = [4 3; 3 2].
只有当两个矩阵的阶数完全相同时，你才能对它们进行加法或减法运算。运算是逐元素进行的：把对应位置的元素相加或相减。例如，若 A = [1 4; −2 0] 且 B = [3 −1; 5 2]，则 A + B = [1+3 4+(−1); −2+5 0+2] = [4 3; 3 2]。
2 5; −7 −2]. Notice that the order of subtraction matters. A − B is not the same as B − A.
减法同理：A − B = [1−3 4−(−1); −2−5 0−2] = [−2 5; −7 −2]。注意减法的顺序很重要。A − B 与 B − A 不相同。
A useful check is that A + B has the same order as A and B. If the orders differ, the sum is undefined, and you should say so rather than forcing an answer.
一个有用的检查是：A + B 的阶数与 A、B 相同。如果阶数不同，和就没有定义，你应该指出这一点，而不是强行计算。
3. Scalar Multiplication | 数乘
element by that scalar. For example, if A = [1 4; −2 0] and k = 3, then 3A = [3 12; −6 0].
用一个数去乘矩阵时，要把矩阵中的每一个元素都乘以这个数。例如，若 A = [1 4; −2 0] 且 k = 3，则 3A = [3 12; −6 0]。
)A = kA + lA and (kl)A = k(lA). These properties are often tested in short exam questions.
数乘对矩阵加法满足分配律：k(A + B) = kA + kB。它还满足 (k + l)A = kA + lA 以及 (kl)A = k(lA)。这些性质经常在短小的考试题中出现。
Combining scalar multiplication with subtraction gives expressions such as 2A − 3B. Work element by element: multiply first, then subtract. Never subtract before multiplying if the question asks for 2A − 3B.
把数乘与减法结合，会得到像 2A − 3B 这样的表达式。要逐元素处理：先乘，再减。如果题目要求 2A − 3B，绝不要先减后乘。
4. Matrix Multiplication | 矩阵乘法
l the number of rows of the second matrix. For 2 × 2 matrices, the product is again 2 × 2.
矩阵乘法不是逐元素相乘。两个矩阵能够相乘的条件是：第一个矩阵的列数等于第二个矩阵的行数。对于 2 × 2 矩阵，乘积仍然是 2 × 2 矩阵。
f+bh; ce+dg cf+dh]. Each entry in the product comes from multiplying a row of A by a column of B and adding the results.
法则是行乘列。若 A = [a b; c d] 且 B = [e f; g h]，则 AB = [ae+bg af+bh; ce+dg cf+dh]。乘积中的每个元素都来自 A 的一行与 B 的一列相乘后求和。
2; 3 4] and B = [5 6; 7 8]. Then AB = [1×5+2×7 1×6+2×8; 3×5+4×7 3×6+4×8] = [19 22; 43 50].
例如，令 A = [1 2; 3 4] 且 B = [5 6; 7 8]。则 AB = [1×5+2×7 1×6+2×8; 3×5+4×7 3×6+4×8] = [19 22; 43 50]。
om AB. This is a common source of exam errors, so always keep the order given in the question.
矩阵乘法一般不可交换：AB 通常不等于 BA。对于上面同样的 A 和 B，BA = [23 34; 31 46]，与 AB 不同。这是考试中常见的错误来源，所以一定要保持题目给出的顺序。
5. Identity and Zero Matrices | 单位矩阵与零矩阵
behaves like the number 1 in ordinary multiplication: for any 2 × 2 matrix A, AI = IA = A.
2 × 2 单位矩阵是 I = [1 0; 0 1]。它像普通乘法中的数字 1 一样：对任何 2 × 2 矩阵 A，都有 AI = IA = A。
O to any matrix leaves it unchanged, and multiplying any matrix by O gives O. Be careful: unlike numbers, two non-zero matrices can multiply to give the zero matrix.
零矩阵是 O = [0 0; 0 0]。任何矩阵加上 O 都不变，任何矩阵乘以 O 都得到 O。要注意：与数字不同，两个非零矩阵相乘可以得到零矩阵。
n A is invertible. Matrix powers follow the usual index laws: AᵐAⁿ = Aᵐ⁺ⁿ and (Aᵐ)ⁿ = Aᵐⁿ.
矩阵的幂由重复乘法定义：A² = AA，A³ = AAA，依此类推。当 A 可逆时，A⁰ 取为 I。矩阵的幂遵循通常的指数律：AᵐAⁿ = Aᵐ⁺ⁿ 以及 (Aᵐ)ⁿ = Aᵐⁿ。
6. Determinant of a 2 × 2 Matrix | 2 × 2 矩阵的行列式
er that tells you important information about the matrix, including whether it has an inverse and how it scales areas.
A = [a b; c d] 的行列式定义为 det A = ad − bc。它是一个数，能告诉你关于该矩阵的重要信息，包括它是否有逆矩阵以及它如何缩放面积。
1 4; 2 8], then det B = 1 × 8 − 4 × 2 = 0. A matrix with determinant 0 is called singular.
例如，若 A = [2 −1; 0 3]，则 det A = 2 × 3 − (−1) × 0 = 6。若 B = [1 4; 2 8]，则 det B = 1 × 8 − 4 × 2 = 0。行列式为 0 的矩阵称为奇异矩阵。
Notation varies: you may see det A, det(A), or |A|. The vertical bars are not absolute value in this context. A non-singular matrix has det A ≠ 0 and is invertible.
记号有多种：你可能看到 det A、det(A) 或 |A|。这里的竖线不是绝对值符号。非奇异矩阵满足 det A ≠ 0，并且是可逆的。
Useful determinant facts: det(AB) = det A × det B, det(Aᵀ A, and det(kA) = k² det A for 2 × 2 matrices. These can save time in multi-part questions.
有用的行列式性质：det(AB) = det A × det B，det(Aᵀ) = det A，对于 2 × 2 矩阵还有 det(kA) = k² det A。这些性质能在多部分题目中节省时间。
7. Inverse of a 2 × 2 Matrix | 2 × 2 矩阵的逆矩阵
⁻¹ = (1/det A) × [d −b; −c a]. Notice that a and d swap places, while b and c change sign.
2 × 2 矩阵 A 的逆矩阵只有当 det A ≠ 0 时才存在。若 A = [a b; c d]，则 A⁻¹ = (1/det A) × [d −b; −c a]。注意 a 与 d 交换位置，而 b 与 c 变号。
1; 0 2] = [1/2 1/6; 0 1/3]. You can check the answer by multiplying: AA⁻¹ should equal I.
对于 A = [2 −1; 0 3]，det A = 6，所以 A⁻¹ = (1/6) × [3 1; 0 2] = [1/2 1/6; 0 1/3]。你可以通过相乘来检验：AA⁻¹ 应该等于 I。
Key properties: (A⁻¹)⁻¹ = A, (AB)⁻¹ = B⁻¹A⁻¹, and (Aᵀ)⁻¹ = (A⁻¹)ᵀ. The reversal of order in (AB)⁻¹ is essential and is frequently examined.
关键性质：(A⁻¹)⁻¹ = A，(AB)⁻¹ = B⁻¹A⁻¹，(Aᵀ)⁻¹ = (A⁻¹)ᵀ。其中 (AB)⁻¹ 中顺序的反转非常重要，经常被考查。
ansformation terms, the matrix collapses the plane onto a line or a point, so information is lost and the transformation cannot be reversed.
若 det A = 0，则 A 没有逆矩阵。用变换的语言说，该矩阵把平面压缩到一条直线或一个点上，信息丢失，因此变换不可逆。
8. Solving Simultaneous Equations | 解联立方程组
oefficient matrix, X = [x; y] is the column of unknowns, and B is the column of constants.
矩阵为求解两个未知数的两个线性方程提供了一种紧凑的方法。把方程组写成 AX = B，其中 A 是系数矩阵，X = [x; y] 是未知数列，B 是常数列。
x form: [2 −1; 0 3][x; y] = [5; 9]. Since det A = 6 ≠ 0, the system has a unique solution.
例如，考虑 2x − y = 5 与 3y = 9。写成矩阵形式：[2 −1; 0 3][x; y] = [5; 9]。因为 det A = 6 ≠ 0，方程组有唯一解。
3], we get X = [1/2×5 + 1/6×9; 0×5 + 1/3×9] = [2.5 + 1.5; 3] = [4; 3]. So x = 4 and y = 3.
两边左乘 A⁻¹：X = A⁻¹B。使用 A⁻¹ = [1/2 1/6; 0 1/3]，得到 X = [1/2×5 + 1/6×9; 0×5 + 1/3×9] = [2.5 + 1.5; 3] = [4; 3]。所以 x = 4，y = 3。
ix method fails; the equations are either inconsistent (no solution) or dependent (infinitely many solutions), and you must analyse them separately.
一定要把解代回原方程检验。若 det A = 0，矩阵法失效；方程组要么不相容（无解），要么相关（无穷多解），必须单独分析。
9. Matrix Transformations in the Plane | 平面中的矩阵变换
c transformation of the plane. A point with position vector [x; y] is mapped to a new point by the product A[x; y]. The origin always maps to the origin.
2 × 2 矩阵可以表示平面的一种几何变换。位置向量为 [x; y] 的点通过乘积 A[x; y] 映射到新的点。原点总是映射到原点。
Common transformations and their matrices are listed below. Angles are measured anticlockwise from the positive x-axis.
常见变换及其矩阵列在下面。角度从 x 轴正方向开始逆时针度量。
- Anticlockwise rotation by θ: [cos θ −sin θ; sin θ cos θ]
- Reflection in the x-axis: [1 0; 0 −1]
- Reflection in the y-axis: [−1 0; 0 1]
- Reflection in the line y = x: [0 1; 1 0]
- Reflection in the line y = −x: [0 −1; −1 0]
- Enlargement with centre O and scale factor k: [k 0; 0 k]
- Stretch parallel to the x-axis with scale factor k: [k 0; 0 1]
- Stretch parallel to the y-axis with scale factor k: [1 0; 0 k]
- Shear parallel to the x-axis with factor k: [1 k; 0 1]
- 逆时针旋转 θ：[cos θ −sin θ; sin θ cos θ]
- 关于 x 轴反射：[1 0; 0 −1]
- 关于 y 轴反射：[−1 0; 0 1]
- 关于直线 y = x 反射：[0 1; 1 0]
- 关于直线 y = −x 反射：[0 −1; −1 0]
- 以原点为中心、比例因子为 k 的放大：[k 0; 0 k]
- 沿 x 方向、比例因子为 k 的拉伸：[k 0; 0 1]
- 沿 y 方向、比例因子为 k 的拉伸：[1 0; 0 k]
- 沿 x 方向、因子为 k 的剪切：[1 k; 0 1]
When two transformations are combined, the matrix of the combined transformation is the product of the individual matrices. If transformation P is followed by transformation Q, the combined matrix is QP, not PQ. Order matters.
当两种变换复合时，复合变换的矩阵是各个矩阵的乘积。如果先做变换 P，再做变换 Q，则复合矩阵是 QP，而不是 PQ。顺序很重要。
10. Determinant as an Area Scale Factor | 行列式作为面积缩放因子
The determinant of a transformation matrix has a geometric meaning: |det A| is the area scale factor. If a shape with area S is transformed by A, the image has area |det A| × S.
变换矩阵的行列式有几何意义：|det A| 是面积缩放因子。若面积为 S 的图形经 A 变换，则像的面积为 |det A| × S。
0 −1; 1 0] has det = 1, so it preserves area; this matches the fact that it is a rotation.
例如，矩阵 [3 0; 0 2] 的行列式为 6，所以它把所有面积乘以 6。矩阵 [0 −1; 1 0] 的行列式为
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