Matrix Algebra for A-Level Maths | A-Level 数学中的矩阵代数

📚 Matrix Algebra for A-Level Maths | A-Level 数学中的矩阵代数

Matrices are a powerful algebraic tool used to organise data, solve systems of linear equations, and describe geometric transformations. In AQA A-Level Mathematics, you need to master the basic operations of matrix algebra, including addition, multiplication, finding determinants and inverses, and applying matrices to real problems.

矩阵是一种强大的代数工具，可以用于整理数据、求解线性方程组以及描述几何变换。在 AQA A-Level 数学中，你需要熟练掌握矩阵代数的基本运算，包括加法、乘法、求行列式与逆矩阵，以及将矩阵应用于实际问题。

1. What is a Matrix? | 什么是矩阵？

A matrix is a rectangular array of numbers arranged in rows and columns. The components inside a matrix are called its elements or entries. We write a matrix inside square brackets, and the size (or order) is given as “rows × columns”. For example, a 2×3 matrix has 2 rows and 3 columns.

矩阵是按行和列排列成矩形阵列的数字。矩阵内部的组成部分称为元素或元。我们用方括号书写矩阵，其大小（或阶数）用“行数 × 列数”表示。例如，一个 2×3 矩阵有 2 行和 3 列。

Here is a general 2×2 matrix:

下面是一个一般的 2×2 矩阵：

A = [ a b ; c d ]

The element in row i and column j is often denoted by aᵢⱼ. For example, in the matrix above, the entry in row 1, column 2 is b.

位于第 i 行第 j 列的元素通常记为 aᵢⱼ。例如，在上面的矩阵中，第 1 行第 2 列的元素是 b。

Two matrices are equal if and only if they have the same order and every corresponding element is equal.

两个矩阵相等，当且仅当它们具有相同的阶数，且每个对应元素都相等。

2. Matrix Addition and Subtraction | 矩阵的加减法

To add or subtract two matrices, they must have exactly the same order. You then add or subtract each corresponding element separately.

两个矩阵进行加法或减法时，它们必须具有完全相同的阶数。然后分别对每个对应元素进行加法或减法运算。

For example, if A = [1 2 ; 3 4] and B = [5 6 ; 7 8], then:

例如，若 A = [1 2 ; 3 4] 且 B = [5 6 ; 7 8]，则：

A + B = [1+5 2+6 ; 3+7 4+8] = [6 8 ; 10 12]

Similarly, A − B = [−4 −4 ; −4 −4].

类似地，A − B = [−4 −4 ; −4 −4]。

Matrix addition is commutative: A + B = B + A. Matrix subtraction is not commutative, because A − B ≠ B − A in general.

矩阵加法满足交换律：A + B = B + A。矩阵减法不满足交换律，因为通常 A − B ≠ B − A。

3. Scalar Multiplication | 标量乘法

A matrix can be multiplied by a scalar (a single ordinary number). To do this, multiply every element of the matrix by that scalar.

矩阵可以与一个标量（即一个普通数值）相乘。做法是将矩阵中的每一个元素都乘以该标量。

If k is a scalar and A = [a b ; c d], then:

如果 k 是一个标量，且 A = [a b ; c d]，则：

kA = [ka kb ; kc kd]

For example, 3 × [1 −2 ; 0 4] = [3 −6 ; 0 12]. Scalar multiplication is distributive over matrix addition: k(A + B) = kA + kB.

例如，3 × [1 −2 ; 0 4] = [3 −6 ; 0 12]。标量乘法对矩阵加法有分配律：k(A + B) = kA + kB。

4. Matrix Multiplication | 矩阵乘法

Matrix multiplication is not carried out element-by-element. Instead, we combine rows of the first matrix with columns of the second matrix using the “row × column” rule. If A is an m×n matrix and B is an n×p matrix, then the product AB is an m×p matrix. The number of columns in A must equal the number of rows in B.

矩阵乘法不是逐元素进行的。相反，我们使用“行 × 列”法则将第一个矩阵的行与第二个矩阵的列结合。若 A 是 m×n 矩阵，B 是 n×p 矩阵，则乘积 AB 是 m×p 矩阵。A 的列数必须等于 B 的行数。

The entry in row i, column j of AB is found by multiplying each element in row i of A by the corresponding element in column j of B, then adding all the products.

AB 的第 i 行第 j 列的元素，是将 A 中第 i 行的每个元素与 B 中第 j 列的对应元素相乘，然后把所有乘积相加得到。

For 2×2 matrices A = [a b ; c d] and B = [e f ; g h]:

对于 2×2 矩阵 A = [a b ; c d] 和 B = [e f ; g h]：

AB = [ae+bg af+bh ; ce+dg cf+dh]

Matrix multiplication is associative: (AB)C = A(BC). However, it is not commutative: in general, AB ≠ BA. Always preserve the order of multiplication.

矩阵乘法满足结合律：(AB)C = A(BC)。但不满足交换律：一般地，AB ≠ BA。始终注意保持矩阵相乘的顺序。

5. The Identity Matrix | 单位矩阵

The identity matrix, denoted I, is a square matrix with 1s on the main diagonal and 0s everywhere else. For 2×2 matrices, the identity matrix is:

单位矩阵记为 I，是一个主对角线上为 1、其余位置为 0 的方阵。对于 2×2 矩阵，单位矩阵为：

I = [1 0 ; 0 1]

The identity matrix acts like the number 1 for matrices. For any compatible matrix A, we have AI = IA = A. Multiplying by the identity matrix leaves the original matrix unchanged.

单位矩阵的作用类似于数值 1。对任意与其相容的矩阵 A，有 AI = IA = A。乘以单位矩阵不会改变原矩阵。

For an n×n matrix, the identity matrix Iₙ has n rows and n columns. In A-Level work, you will usually use the 2×2 identity matrix.

对于 n×n 矩阵，单位矩阵 Iₙ 有 n 行 n 列。在 A-Level 中，通常使用 2×2 单位矩阵。

6. Determinant of a 2×2 Matrix | 二阶矩阵的行列式

The determinant is a scalar value associated with a square matrix. For a 2×2 matrix A = [a b ; c d], the determinant is written as det(A) or |A| and is calculated as:

行列式是与一个方阵相关的标量值。对于 2×2 矩阵 A = [a b ; c d]，行列式记作 det(A) 或 |A|，其计算方式为：

det(A) = ad − bc

If the determinant is zero, the matrix is called singular. A singular matrix has no inverse. A nonzero determinant means the matrix is non-singular (invertible).

如果行列式为零，则该矩阵称为奇异矩阵。奇异矩阵没有逆矩阵。行列式非零意味着矩阵是非奇异的（可逆的）。

The determinant also gives the area scale factor of a linear transformation. A determinant of −2 means the area is doubled and the orientation is reversed.

行列式还给出了线性变换的面积缩放因子。行列式为 −2 意味着面积变为原来的两倍，且方向反转。

7. Inverse of a 2×2 Matrix | 二阶矩阵的逆矩阵

The inverse of a matrix A is denoted A⁻¹. It satisfies AA⁻¹ = A⁻¹A = I. Only square matrices with a non-zero determinant have an inverse.

矩阵 A 的逆矩阵记为 A⁻¹。它满足 AA⁻¹ = A⁻¹A = I。只有行列式不为零的方阵才有逆矩阵。

For A = [a b ; c d], the inverse is given by:

对于 A = [a b ; c d]，其逆矩阵为：

A⁻¹ = 1/(ad−bc) × [d −b ; −c a]

Note the pattern: swap the diagonal elements a and d, change the signs of b and c, then divide by the determinant ad − bc.

注意规律：交换主对角线元素 a 和 d，改变 b 和 c 的符号，再除以行列式 ad − bc。

For example, if A = [2 1 ; 5 3], then det(A) = 6 − 5 = 1, so A⁻¹ = [3 −1 ; −5 2]. You can check that A × A⁻¹ = I.

例如，若 A = [2 1 ; 5 3]，则 det(A) = 6 − 5 = 1，所以 A⁻¹ = [3 −1 ; −5 2]。你可以验证 A × A⁻¹ = I。

8. Solving Simultaneous Equations with Matrices | 用矩阵求解联立方程

Matrices provide a systematic method for solving pairs of simultaneous linear equations. Consider the equations:

矩阵为求解一对线性联立方程提供了一种系统化方法。考虑方程组：

ax + by = e
cx + dy = f

Write the system in matrix form as:

将方程组写成矩阵形式：

[a b ; c d] [x ; y] = [e ; f]

Let A = [a b ; c d], X = [x ; y] and B = [e ; f]. Then AX = B. If det(A) ≠ 0, multiply both sides on the left by A⁻¹:

设 A = [a b ; c d]，X = [x ; y]，B = [e ; f]。则 AX = B。若 det(A) ≠ 0，则等式两边左乘 A⁻¹：

X = A⁻¹B

Then multiply the inverse matrix by the column vector B to obtain x and y. If det(A) = 0, the system may have no solution or infinitely many solutions.

然后将逆矩阵乘以列向量 B，即可得到 x 和 y。如果 det(A) = 0，则方程组可能无解或有无穷多组解。

It is important to multiply on the left by A⁻¹, not on the right, because matrix multiplication is not commutative.

重要的是必须左乘 A⁻¹，而不是右乘，因为矩阵乘法不满足交换律。

9. Transformation Matrices | 变换矩阵

A 2×2 matrix can represent a linear transformation in the plane. When a column vector [x ; y] is multiplied by a transformation matrix, the result is the image of the point under that transformation.

一个 2×2 矩阵可以表示平面中的线性变换。当列向量 [x ; y] 乘以一个变换矩阵时，结果就是该点在该变换下的像。

Common transformation matrices include:

常见的变换矩阵包括：

Reflection in the x-axis: [1 0 ; 0 −1]
反射关于 x 轴：[1 0 ; 0 −1]
Reflection in the y-axis: [−1 0 ; 0 1]
反射关于 y 轴：[−1 0 ; 0 1]
Reflection in the line y = x: [0 1 ; 1 0]
反射关于直线 y = x：[0 1 ; 1 0]
Rotation about the origin by angle θ anticlockwise: [cosθ −sinθ ; sinθ cosθ]
绕原点逆时针旋转角度 θ：[cosθ −sinθ ; sinθ cosθ]
Enlargement with scale factor k: [k 0 ; 0 k]
以原点为中心、缩放因子为 k 的伸缩变换：[k 0 ; 0 k]
Shear in the x-direction: [1 k ; 0 1]
沿 x 方向的切变：[1 k ; 0 1]

To find the image of a point, multiply the transformation matrix by the column vector of the point’s coordinates.

要求一个点的像，只需用变换矩阵乘以该点坐标的列向量。

10. Composition of Transformations | 变换的复合

If two transformations are applied in succession, the combined effect can be represented by a single matrix found by multiplying the individual transformation matrices. Important: the transformation applied first must be on the right, and the one applied second must be on the left.

如果依次施加两个变换，其复合效果可由单个矩阵表示，该矩阵通过将各个变换矩阵相乘来得到。重要：先进行的变换放在右侧，后进行的变换放在左侧。

For example, if transformation P is applied first, followed by Q, then the combined matrix is QP, not PQ. This is because we denote a vector v, then apply P to get Pv, then apply Q to get Q(Pv) = (QP)v.

例如，若先进行变换 P，然后再进行 Q，则复合矩阵为 QP，而不是 PQ。这是因为对于向量 v，先施加 P 得到 Pv，再施加 Q 得到 Q(Pv) = (QP)v。

When composing transformations, always be careful with the order. A common mistake is to reverse the matrices. Check the effect on a simple point such as (1,0) to verify your order.

复合变换时，务必注意顺序。一个常见错误是将矩阵的前后顺序搞反。可以通过对一个简单点（如 (1,0)）的效果来验证顺序。

Also remember that the determinant of the product is the product of the determinants: det(QP) = det(Q) det(P). This can be useful for checking the area scale factor of a combined transformation.

同时要记得，乘积的行列式等于行列式的乘积：det(QP) = det(Q) det(P)。这对于检查复合变换的面积缩放因子很有用。

11. Practical Tips and Common Mistakes | 实践技巧与常见错误

Here are some key points to remember when working with matrix algebra in exams:

以下是在考试中进行矩阵代数运算时需记住的一些要点：

Always check that matrices have compatible orders before multiplication. If this condition fails, the product is undefined.
在进行乘法前，务必检查矩阵的阶数是否相容。若条件不满足，乘积无定义。
When writing an inverse, do not forget the factor 1/det(A). Many students compute the adjugate matrix correctly but forget to divide by the determinant.
写出逆矩阵时，不要忘记因子 1/det(A)。许多学生能正确算出伴随矩阵，却忘记除以行列式。
If det(A) = 0, the matrix is singular and has no inverse. Do not attempt to calculate one.
若 det(A) = 0，矩阵是奇异的，没有逆矩阵。不要尝试去计算逆矩阵。
Always multiply both sides of a matrix equation by the inverse on the same side (usually the left).
在矩阵方程两边乘以逆矩阵时，必须保持在同一侧（通常是左侧）进行乘法。
When identifying a transformation from a matrix, test the effect on the points (1,0) and (0,1) rather than trying to memorise every result.
从矩阵识别变换时，可以通过考察点 (1,0) 和 (0,1) 的效果来判断，而不必试图记住所有结果。
For rotation matrices, be sure your calculator is in the correct angle mode (radians or degrees) when finding exact values.
对于旋转矩阵，在计算精确值时，请确保计算器处于正确的角度模式（弧度或角度）。

Practice with past papers is essential because matrix algebra questions often combine several skills, such as solving equations, calculating determinants, and interpreting transformations.

通过真题进行练习尤为重要，因为矩阵代数题目通常综合了多项技能，例如求解方程、计算行列式以及理解变换。

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Matrix Algebra for A-Level Maths | A-Level 数学中的矩阵代数