📚 Matrices: Operations, Transformations and Systems of Equations | 矩阵：运算、变换与方程组
Matrices are rectangular arrays of numbers used to represent data, linear transformations, and coefficients in simultaneous equations. In AQA A-Level Mathematics, matrix methods are essential for solving systems of equations, describing geometric transformations, and understanding invariant properties.
矩阵是由数字组成的矩形阵列，用于表示数据、线性变换以及联立方程中的系数。在 AQA A-Level 数学中，矩阵方法是解方程组、描述几何变换和理解不变性质的重要工具。
1. What is a Matrix? | 矩阵是什么
3 matrix has 2 rows and 3 columns. Each element is identified by its position, such as aᵢⱼ for the element in row i and column j.
矩阵是一个按行和列排列的矩形元素阵列。它的阶数用行数 × 列数表示，因此 2 × 3 矩阵有 2 行和 3 列。每个元素由其位置确定，例如 aᵢⱼ 表示第 i 行第 j 列的元素。
A = [aᵢⱼ] means A is an m × n matrix with entries aᵢⱼ.
A = [aᵢⱼ] 表示 A 是一个 m × n 矩阵，其元素为 aᵢⱼ。
2. Matrix Addition and Scalar Multiplication | 矩阵加法与数乘
Two matrices can be added only if they have the same order. Addition is performed element by element: corresponding entries are added together. Scalar multiplication multiplies every element of the matrix by the same scalar.
两个矩阵只有在阶数相同时才能相加。加法按元素逐个进行：对应元素相加。数乘则是将矩阵中的每个元素都乘以同一个标量。
(A + B)ᵢⱼ = aᵢⱼ + bᵢⱼ and (kA)ᵢⱼ = k aᵢⱼ
(A + B)ᵢⱼ = aᵢⱼ + bᵢⱼ 且 (kA)ᵢⱼ = k aᵢⱼ
ple, if A = [1 2; 3 4] and B = [5 6; 7 8], then A + B = [6 8; 10 12] and 3A = [3 6; 9 12].
例如，若 A = [1 2; 3 4]、B = [5 6; 7 8]，则 A + B = [6 8; 10 12]，3A = [3 6; 9 12]。
3. Matrix Multiplication | 矩阵乘法
For the product AB to exist, the number of columns in A must equal the number of rows in B. The element in row i and column j of AB is found by multiplying row i of A by column j of B and adding the products.
要使乘积 AB 存在，A 的列数必须等于 B 的行数。AB 中第 i 行第 j 列的元素由 A 的第 i 行与 B 的第 j 列对应元素相乘后求和得到。
(AB)ᵢⱼ = Σ aᵢₖ bₖⱼ
(AB)ᵢⱼ = Σ aᵢₖ bₖⱼ
2; 3 4] × [5 6; 7 8] = [19 22; 43 50], but reversing the order gives a different product.
矩阵乘法一般不满足交换律，因此 AB 通常不等于 BA。例如，[1 2; 3 4] × [5 6; 7 8] = [19 22; 43 50]，但交换顺序后得到的结果不同。
4. The Identity Matrix and Zero Matrix | 单位矩阵与零矩阵
= A. The zero matrix O has every element equal to zero and acts as the additive identity.
单位矩阵 I 是一个主对角线为 1、其余元素为 0 的方阵。对于任何可相乘的矩阵 A，都有 IA = AI = A。零矩阵 O 的每个元素都为 0，它在加法中起单位元的作用。
I₂ = [1 0; 0 1] and I₃ = [1 0 0; 0 1 0; 0 0 1]
I₂ = [1 0; 0 1] 且 I₃ = [1 0 0; 0 1 0; 0 0 1]
5. Determinants of 2 × 2 and 3 × 3 Matrices | 2 × 2 和 3 × 3 矩阵的行列式
rse. For a 3 × 3 matrix, the determinant is calculated by expanding along a row or column.
对于 2 × 2 矩阵 A = [a b; c d]，其行列式为 det(A) = ad − bc。如果 det(A) = 0，则该矩阵为奇异矩阵，不存在逆矩阵。对于 3 × 3 矩阵，行列式沿某一行或某一列展开计算。
det [a b; c d] = ad − bc
det [a b; c d] = ad − bc
h i], the expansion along the first row is det(A) = a(ei − fh) − b(di − fg) + c(dh − eg).
对于 3 × 3 矩阵 A = [a b c; d e f; g h i]，沿第一行展开为 det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)。
6. The Inverse of a Matrix | 矩阵的逆
se is found by swapping the leading diagonal entries, changing the signs of the other two entries, and dividing by the determinant.
如果方阵 A 满足 det(A) ≠ 0，则它是可逆的。逆矩阵 A⁻¹ 满足 AA⁻¹ = A⁻¹A = I。对于 2 × 2 矩阵，求逆的方法是交换主对角线元素、改变另外两个元素的符号，并除以行列式。
If A = [a b; c d], then A⁻¹ = 1/(ad − bc) [d −b; −c a].
若 A = [a b; c d]，则 A⁻¹ = 1/(ad − bc) [d −b; −c a]。
nverse of a product reverses the order: (AB)⁻¹ = B⁻¹A⁻¹. This is a very common exam point.
乘积的逆会交换顺序：(AB)⁻¹ = B⁻¹A⁻¹。这是考试中非常常见的考点。
7. Solving Simultaneous Equations with Matrices | 用矩阵解联立方程
is x = A⁻¹b. This method is particularly efficient for systems with two or three unknowns.
联立方程组可以写成单个矩阵方程 Ax = b。如果 A 可逆，则唯一解为 x = A⁻¹b。对于含有两个或三个未知数的方程组，这种方法尤其高效。
]. Since det = 2 × 1 − 3 × 4 = −10 ≠ 0, the solution is [x; y] = A⁻¹[5; 6] = [13/10; 4/5].
例如，方程组 2x + 3y = 5 和 4x + y = 6 可写为 [2 3; 4 1][x; y] = [5; 6]。由于 det = 2 × 1 − 3 × 4 = −10 ≠ 0，解为 [x; y] = A⁻¹[5; 6] = [13/10; 4/5]。
8. Geometric Transformations in 2D | 二维几何变换
ansformation of the plane. The columns of the matrix are the images of the unit basis vectors (1,0) and (0,1). This makes it easy to identify the transformation from its matrix.
2 × 2 矩阵可以表示平面上的线性变换。矩阵的列是单位基向量 (1,0) 和 (0,1) 的像。因此从矩阵识别变换非常容易。
| Transformation / 变换 | Matrix / 矩阵 |
| 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] |
| Rotation by θ anticlockwise / 逆时针旋转 θ | [cos θ −sin θ; sin θ cos θ] |
| Stretch factor k in x-direction / x 方向缩放因子 k | [k 0; 0 1] |
| Shear parallel to x-axis factor k / 平行 x 轴剪切因子 k | [1 k; 0 1] |
9. Combining Transformations | 变换的组合
When two transformations are combined, their matrices are multiplied. The product BA means apply A first, then apply B. The order is essential because matrix multiplication is not commutative, and reversing the order usually gives a different transformation.
两个变换组合时，其矩阵相乘。乘积 BA 表示先施加 A，再施加 B。顺序非常重要，因为矩阵乘法不满足交换律，交换顺序通常会得到不同的变换。
everse the combined transformation, use the inverse in the reverse order: (BA)⁻¹ = A⁻¹B⁻¹.
如果先做变换 A，再做变换 B，组合矩阵为 BA。要反转组合变换，按相反顺序使用逆矩阵：(BA)⁻¹ = A⁻¹B⁻¹。
10. Invariant Points and Lines | 不变点与不变线
n be solved to find all invariant points. An invariant line is a line whose points map to points on the same line, even if individual points move along it.
如果位置向量保持不变，即 Mx = x，则该点在矩阵变换下是不变点。解此方程可找到所有不变点。不变线是指线上任意点变换后仍在同一条直线上的直线，即使个别点沿直线移动。
Invariant lines often require solving a matrix equation with an unknown scalar multiplier.
例如，在关于 x 轴反射的变换下，x 轴本身是不变点所在直线，因为其上每个点都满足 Mx = x。求不变线通常需要解含有未知标量乘子的矩阵方程。
11. Exam Tips and Common Mistakes | 考试技巧与常见错误
Always check that matrix multiplication is possible before carrying it out: the number of columns in the first matrix must match the number of rows in the second. When finding an inverse, first calculate the determinant and confirm it is non-zero.
进行矩阵乘法前一定要检查是否可行：第一个矩阵的列数必须等于第二个矩阵的行数。求逆矩阵时，先计算行列式并确认其不为零。
ansformation questions, pay close attention to the order of multiplication. For simultaneous equations, check whether the coefficient matrix is singular before writing the solution.
不要假设 AB = BA。在逆矩阵和变换问题中，要特别注意乘法顺序。对于联立方程，在写出解之前先检查系数矩阵是否奇异。
det(A) = 0 means no unique inverse and no unique solution to Ax = b.
det(A) = 0 意味着没有唯一逆矩阵，Ax = b 也没有唯一解。
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