Matrices | 矩阵

📚 Matrices | 矩阵

Matrices are rectangular arrays of numbers that form a cornerstone of A-Level Mathematics. They enable us to solve systems of linear equations efficiently, represent geometric transformations, and organise data in a structured way. In this guide, we cover all matrix topics required by the AQA A-Level specification, with worked examples and exam-style insights.

矩阵是数字的矩形阵列，是A-Level数学的重要基石。它们使我们能够高效地求解线性方程组、表示几何变换并以结构化的方式组织数据。在本指南中，我们将涵盖AQA A-Level考试大纲所要求的全部矩阵专题，配有例题和考试风格的要点解析。

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1. What Is a Matrix | 什么是矩阵

A matrix is a rectangular arrangement of numbers (called elements or entries) enclosed in square brackets. Its size or order is described as ‘rows × columns’. For example, a 2 × 3 matrix has 2 rows and 3 columns.

矩阵是将数字（称为元素或元）排列成矩形并用方括号括起来的结构。其大小或阶数用”行 × 列”来描述。例如，一个2 × 3矩阵有2行和3列。

In AQA A-Level Mathematics, the most frequently encountered matrices are 2 × 2 matrices, which take the form:

A = [ a b ; c d ]

Here, the rows run horizontally and the columns vertically. The element in the iᵗʰ row and jᵗʰ column is often denoted aᵢⱼ, where the first subscript is the row number and the second is the column number.

这里，行是水平方向的，列是垂直方向的。位于第 i 行和第 j 列的元素通常记为 aᵢⱼ，其中第一个下标是行号，第二个下标是列号。

For instance, in the matrix above, the element b is in row 1, column 2, so we write a₁₂ = b. The element c is in row 2, column 1, so a₂₁ = c. Understanding this notation is essential for advanced work such as computing products of matrices.

例如，在上面的矩阵中，元素 b 位于第1行第2列，因此我们记 a₁₂ = b。元素 c 位于第2行第1列，因此 a₂₁ = c。理解这种记法对于后续计算矩阵乘积非常重要。

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2. Special Matrices | 特殊矩阵

Several types of matrices appear repeatedly in A-Level problems, and you should recognise them instantly.

有几类矩阵在A-Level题目中反复出现，你应该能够一眼认出它们。

The zero matrix is a matrix in which every element is 0. It acts as the additive identity: adding the zero matrix to any matrix of the same order leaves the original matrix unchanged.

零矩阵是所有元素都为0的矩阵。它充当加法单位元：将零矩阵与任何同阶矩阵相加，原矩阵保持不变。

The identity matrix, denoted I, is a square matrix with 1s on the main diagonal (from top-left to bottom-right) and 0s elsewhere. For 2 × 2 matrices:

单位矩阵记为 I，是一个方阵，主对角线（从左上到右下）上的元素为1，其余位置为0。对于2 × 2矩阵：

I = [ 1 0 ; 0 1 ]

The identity matrix acts as the multiplicative identity: for any 2 × 2 matrix A, we have AI = IA = A. It plays a role in matrix algebra analogous to the number 1 in ordinary arithmetic.

单位矩阵充当乘法单位元：对于任意2 × 2矩阵A，都有 AI = IA = A。它在矩阵代数中的作用类似于数字1在普通算术中的作用。

Finally, a singular matrix is a square matrix whose determinant is zero. As we shall see, singular matrices do not have an inverse and correspond to systems of equations with no unique solution.

最后，奇异矩阵是行列式为零的方阵。我们稍后将看到，奇异矩阵没有逆矩阵，对应着没有唯一解的方程组。

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3. Addition and Scalar Multiplication | 加法与数乘

Two matrices of the same order can be added (or subtracted) by adding (or subtracting) corresponding elements. If the orders differ, the operation is undefined.

两个同阶矩阵可以相加（或相减），只需将对应元素相加（或相减）。如果阶数不同，则该运算无定义。

For example, suppose A = [ 1 3 ; 2 0 ] and B = [ 4 −1 ; 5 2 ]. Then:

例如，设 A = [ 1 3 ; 2 0 ]，B = [ 4 −1 ; 5 2 ]，则：

A + B = [ 1+4 3+(−1) ; 2+5 0+2 ] = [ 5 2 ; 7 2 ]

The addition is performed element-wise, and the order of addition does not matter: A + B = B + A. This property, known as commutativity, mirrors ordinary number arithmetic.

加法是逐元素进行的，加法的顺序不影响结果：A + B = B + A。这个性质称为交换律，与普通数字运算一致。

Scalar multiplication involves multiplying every element of a matrix by a scalar (a single number). If k is a scalar and A = [ a b ; c d ], then:

数乘是用一个标量（单个数字）乘以矩阵中的每一个元素。如果 k 是标量，A = [ a b ; c d ]，则：

kA = [ ka kb ; kc kd ]

For instance, 3A = [ 3 9 ; 6 0 ] using the matrix A above. Note that division by a scalar is simply multiplication by its reciprocal: A ÷ 2 = ½A.

例如，使用上面的矩阵A，3A = [ 3 9 ; 6 0 ]。注意除以一个标量就是乘以它的倒数：A ÷ 2 = ½A。

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4. Matrix Multiplication | 矩阵乘法

Matrix multiplication is more subtle than addition. For two matrices to be multiplied, the number of columns of the first matrix must equal the number of rows of the second matrix. The resulting matrix has the number of rows of the first matrix and the number of columns of the second.

矩阵乘法比加法更为微妙。两个矩阵相乘的前提是，第一个矩阵的列数必须等于第二个矩阵的行数。结果矩阵的行数等于第一个矩阵的行数，列数等于第二个矩阵的列数。

For two 2 × 2 matrices A = [ a₁ b₁ ; c₁ d₁ ] and B = [ a₂ b₂ ; c₂ d₂ ], the product AB is defined as:

对于两个2 × 2矩阵 A = [ a₁ b₁ ; c₁ d₁ ] 和 B = [ a₂ b₂ ; c₂ d₂ ]，乘积 AB 定义为：

AB = [ a₁a₂ + b₁c₂ a₁b₂ + b₁d₂ ; c₁a₂ + d₁c₂ c₁b₂ + d₁d₂ ]

Each entry in the product is obtained by multiplying the elements of a row of the first matrix by the corresponding elements of a column of the second matrix, and then summing the products. This is often described as the ‘row × column’ rule.

乘积中的每个元素都是通过将第一个矩阵某一行的元素与第二个矩阵某一列的对应元素相乘，然后将乘积求和得到的。这通常被称为”行 × 列”法则。

Let us work through a concrete example. If A = [ 1 2 ; 3 4 ] and B = [ 5 6 ; 7 8 ], then:

让我们看一个具体例子。若 A = [ 1 2 ; 3 4 ]，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 ]

A crucial fact is that matrix multiplication is not commutative. In general, AB ≠ BA. You should always preserve the order of multiplication. Indeed, sometimes AB is defined while BA is not defined, or both are defined but give different results.

一个关键事实是矩阵乘法不满足交换律。一般地，AB ≠ BA。你应当始终保持乘法的顺序。事实上，有时 AB 有定义而 BA 无定义，或者两者都有定义但结果不同。

However, matrix multiplication is associative: (AB)C = A(BC). It is also distributive over addition: A(B + C) = AB + AC. These algebraic laws allow us to manipulate matrix expressions systematically.

然而，矩阵乘法满足结合律：(AB)C = A(BC)。它还对加法满足分配律：A(B + C) = AB + AC。这些代数法则使我们能够系统地化简矩阵表达式。

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5. Determinant of a 2 × 2 Matrix | 2 × 2矩阵的行列式

The determinant is a scalar value associated with a square matrix. For the 2 × 2 matrix A = [ a b ; c d ], the determinant is:

行列式是与方阵相关的一个标量值。对于2 × 2矩阵 A = [ a b ; c d ]，其行列式为：

det(A) = ad − bc

The determinant is often written using vertical bars: |A|. For example, if A = [ 2 −1 ; 4 3 ], then det(A) = (2)(3) − (−1)(4) = 6 + 4 = 10.

行列式常用竖线表示：|A|。例如，若 A = [ 2 −1 ; 4 3 ]，则 det(A) = (2)(3) − (−1)(4) = 6 + 4 = 10。

If det(A) = 0, the matrix is said to be singular. A singular matrix has no inverse, and when used to represent a system of linear equations, the system either has no solutions or infinitely many solutions. If det(A) ≠ 0, the matrix is invertible (non-singular).

若 det(A) = 0，则该矩阵称为奇异矩阵。奇异矩阵没有逆矩阵，当它用于表示线性方程组时，方程组要么无解，要么有无穷多解。若 det(A) ≠ 0，则该矩阵是可逆的（非奇异的）。

Geometrically, the determinant of a transformation matrix represents the scale factor of the area change. A determinant of −2, for instance, means the area is doubled but the orientation of the shape is reversed (reflected). A zero determinant means the transformation collapses the plane into a line or a point.

从几何角度看，变换矩阵的行列式表示面积变化的比例因子。例如，行列式为 −2 意味着面积变为原来的2倍，但图形方向被翻转（反射）。行列式为零意味着变换将平面压缩为一条线或一个点。

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6. Inverse of a 2 × 2 Matrix | 2 × 2矩阵的逆

The inverse of a matrix A, denoted A⁻¹, is the matrix that satisfies A⁻¹A = AA⁻¹ = I, where I is the identity matrix. For a 2 × 2 matrix A = [ a b ; c d ], the inverse is:

矩阵 A 的逆矩阵记为 A⁻¹，它满足 A⁻¹A = AA⁻¹ = I，其中 I 是单位矩阵。对于2 × 2矩阵 A = [ a b ; c d ]，其逆矩阵为：

A⁻¹ = (1 / (ad − bc)) × [ d −b ; −c a ]

Notice that the elements on the main diagonal (a and d) are swapped, the signs of the off-diagonal elements (b and c) are reversed, and the entire matrix is multiplied by the reciprocal of the determinant. If ad − bc = 0, the inverse does not exist.

注意，主对角线上的元素（a和d）互换位置，非对角线元素（b和c）改变符号，然后整个矩阵乘以行列式的倒数。若 ad − bc = 0，则逆矩阵不存在。

Let us find the inverse of A = [ 4 1 ; 3 2 ]. First, det(A) = (4)(2) − (1)(3) = 8 − 3 = 5. Then:

让我们求 A = [ 4 1 ; 3 2 ] 的逆矩阵。首先，det(A) = (4)(2) − (1)(3) = 8 − 3 = 5。然后：

A⁻¹ = (1/5) × [ 2 −1 ; −3 4 ] = [ 2/5 −1/5 ; −3/5 4/5 ]

You should always verify your answer by checking that AA⁻¹ = I. This is a quick and reliable way to catch arithmetic errors in the exam.

你应该始终通过检验 AA⁻¹ = I 来验证答案。这是一种快速而可靠的方法，可以帮助你在考试中捕捉计算错误。

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7. Solving Simultaneous Equations | 解联立方程组

One of the most important applications of matrices is solving simultaneous linear equations. Consider the system:

矩阵最重要的应用之一就是求解联立线性方程组。考虑如下方程组：

ax + by = p
cx + dy = q

This system can be written in matrix form as:

这个方程组可以写成矩阵形式：

[ a b ; c d ] [ x ; y ] = [ p ; q ]

Denoting the coefficient matrix by A, the variable matrix by X and the constant matrix by B, we have AX = B. To solve for X, multiply both sides on the left by A⁻¹:

将系数矩阵记为 A，变量矩阵记为 X，常数矩阵记为 B，则有 AX = B。为了求解 X，在等式两边同时左乘 A⁻¹：

X = A⁻¹B

Let us apply this to the system 2x + 3y = 7 and 4x + 5y = 13. In matrix form:

我们将此方法应用于方程组 2x + 3y = 7 和 4x + 5y = 13。其矩阵形式为：

[ 2 3 ; 4 5 ] [ x ; y ] = [ 7 ; 13 ]

Here, det(A) = (2)(5) − (3)(4) = 10 − 12 = −2. Therefore A⁻¹ = (−1/2) × [ 5 −3 ; −4 2 ]. Multiplying by B gives:

这里，det(A) = (2)(5) − (3)(4) = 10 − 12 = −2。因此 A⁻¹ = (−1/2) × [ 5 −3 ; −4 2 ]。将其与 B 相乘得到：

[ x ; y ] = (−1/2) × [ 5×7 + (−3)×13 ; (−4)×7 + 2×13 ] = (−1/2) × [ −4 ; −2 ] = [ 2 ; 1 ]

Hence x = 2 and y = 1. You should always substitute the solutions back into the original equations to confirm they are correct.

因此 x = 2，y = 1。你应该始终将解代回原方程以确认其正确性。

If det(A) = 0, the matrix A is singular and A⁻¹ does not exist. In this case, the two lines represented by the equations are parallel. They may be distinct (no solutions) or coincident (infinitely many solutions). You can distinguish these cases by comparing the equations: if one equation is a multiple of the other, the lines coincide.

若 det(A) = 0，则矩阵 A 是奇异的，A⁻¹ 不存在。此时，方程所代表的两条直线平行。它们可能是不同的直线（无解）或重合的直线（无穷多解）。你可以通过比较方程来区分这两种情况：若一个方程是另一个方程的倍数，则两直线重合。

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8. Geometric Transformations | 几何变换

A 2 × 2 matrix can represent a linear transformation of the plane. If point P has position vector [ x ; y ], then its image P’ under the transformation matrix A is given by:

一个2 × 2矩阵可以表示平面的线性变换。若点 P 的位置向量为 [ x ; y ]，则其在变换矩阵 A 作用下的像 P’ 为：

[ x’ ; y’ ] = A [ x ; y ]

Several standard transformation matrices are particularly important for the AQA specification. The identity matrix I = [ 1 0 ; 0 1 ] leaves every point unchanged, representing the identity transformation.

对于AQA考试大纲，有若干标准变换矩阵尤为重要。单位矩阵 I = [ 1 0 ; 0 1 ] 使每个点保持不变，代表恒等变换。

Reflection in the x-axis is given by [ 1 0 ; 0 −1 ], while reflection in the y-axis uses [ −1 0 ; 0 1 ]. Reflection in the line y = x is represented by [ 0 1 ; 1 0 ].

反射变换方面，关于x轴的反射为 [ 1 0 ; 0 −1 ]，关于y轴的反射为 [ −1 0 ; 0 1 ]，关于直线 y = x 的反射为 [ 0 1 ; 1 0 ]。

Rotation anticlockwise about the origin through angle θ is given by:

绕原点逆时针旋转角度 θ 的变换矩阵为：

R(θ) = [ cosθ −sinθ ; sinθ cosθ ]

For example, rotation by 90° anticlockwise uses cos 90° = 0 and sin 90° = 1, giving [ 0 −1 ; 1 0 ]. Rotation by 180° gives [ −1 0 ; 0 −1 ], which is equivalent to a half-turn.

例如，逆时针旋转90°时，cos 90° = 0，sin 90° = 1，因此矩阵为 [ 0 −1 ; 1 0 ]。旋转180°时矩阵为 [ −1 0 ; 0 −1 ]，相当于旋转半圈。

Enlargement with scale factor k centred at the origin is represented by [ k 0 ; 0 k ]. A non-uniform scaling, sometimes called a stretch, is represented by a diagonal matrix such as [ s 0 ; 0 1 ], which stretches by factor s in the x-direction.

以原点为中心的放大变换，比例因子为 k，矩阵为 [ k 0 ; 0 k ]。非均匀缩放（有时称为拉伸）用对角矩阵表示，例如 [ s 0 ; 0 1 ] 表示在x方向上拉伸 s 倍。

Combined transformations are handled by matrix multiplication. If transformation A is applied first, followed by transformation B, the combined transformation matrix is BA. This order is critical: the transformation applied first is written on the right in the product. For example, a rotation by 90° followed by a reflection in the x-axis has matrix:

复合变换通过矩阵乘法处理。如果先施加变换 A，再施加变换 B，则复合变换矩阵为 BA。这个顺序至关重要：先施加的变换写在乘积的右边。例如，先旋转90°再关于x轴反射，其复合矩阵为：

M = [ 1 0 ; 0 −1 ] × [ 0 −1 ; 1 0 ] = [ 0 −1 ; −1 0 ]

Geometrically, the determinant of a transformation matrix tells us the area scale factor, and a negative determinant indicates that the orientation has been reversed (a reflection is involved). An enlargement with scale factor k has determinant k², matching the area ratio.

从几何角度看，变换矩阵的行列式告诉我们面积的比例因子，而行列式为负表示方向被反转（涉及反射）。比例因子为 k 的放大变换，其行列式为 k²，与面积比一致。

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9. Exam Tips and Common Pitfalls | 考试技巧与常见陷阱

Students frequently lose marks on matrix questions due to avoidable errors. The most common mistake is reversing the order in matrix multiplication. Always check that the row from the first matrix is paired with the column from the second matrix in the correct sequence.

学生在矩阵题目中经常因为可以避免的错误而失分。最常见的错误是在矩阵乘法中弄反顺序。始终检查第一个矩阵的行与第二个矩阵的列是否按正确顺序配对。

When computing an inverse, be meticulous with signs. The off-diagonal elements b and c change sign, while the diagonal elements a and d are swapped. It is easy to forget the negative sign or the reciprocal factor, so verify using AA⁻¹ = I.

在计算逆矩阵时，要非常细心处理符号。非对角线元素 b 和 c 要变号，而对角线元素 a 和 d 要互换位置。很容易忘记负号或倒数因子，因此务必用 AA⁻¹ = I 进行验证。

For transformation problems, always check whether the question specifies ‘anticlockwise’ or ‘clockwise’. A clockwise rotation through angle θ uses the matrix [ cosθ sinθ ; −sinθ cosθ ], which is the inverse of the anticlockwise rotation matrix.

对于变换问题，始终检查题目中指定的是”逆时针”还是”顺时针”。顺时针旋转角度 θ 的矩阵为 [ cosθ sinθ ; −sinθ cosθ ]，它是逆时针旋转矩阵的逆矩阵。

Finally, remember that for a system of equations to have a unique solution, the determinant of the coefficient matrix must be non-zero. If det(A) = 0, do not attempt to find A⁻¹ — instead, analyse the equations to determine whether there are no solutions or infinitely many.

最后，记住方程组有唯一解的条件是系数矩阵的行列式不为零。若 det(A) = 0，不要试图求 A⁻¹——而应分析方程本身，判断是无解还是有无穷多解。

Mastering matrices requires practice with both algebraic manipulation and geometric interpretation. Work through past paper questions systematically, and always write out each step clearly to avoid careless errors.

掌握矩阵需要同时在代数运算和几何解释两方面下功夫。系统性地练习历年真题，并且始终清晰地写出每一步，以尽量避免粗心错误。

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