Matrices | 矩阵

📚 Matrices | 矩阵

Matrices are one of the most powerful tools in A-level Mathematics. They allow us to organise numbers, solve simultaneous equations, and describe geometric transformations in a compact and efficient way. In this revision guide, we will cover everything you need to know for your AQA A-level exam: notation, addition, subtraction, multiplication, determinants, inverses, and applications to transformations and equations.

矩阵是 A-level 数学中最强大的工具之一。它使我们能够以紧凑高效的方式组织数字、求解联立方程组以及描述几何变换。在本复习指南中，我们将覆盖 AQA A-level 考试所需的全部内容：记号、加法、减法、乘法、行列式、逆矩阵，以及它们在变换和方程组中的应用。

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1. What is a Matrix? | 什么是矩阵？

A matrix is a rectangular array of numbers arranged in rows and columns. We usually denote a matrix with a capital letter, for example (A), and we write its size as “rows × columns”. A matrix with (m) rows and (n) columns is called an (m times n) matrix.

矩阵是按行和列排列成矩形阵列的数字。我们通常用大写字母表示矩阵，例如 (A)，并将其大小写为“行数 × 列数”。一个有 (m) 行 (n) 列的矩阵称为 (m times n) 矩阵。

For example, the matrix

A = [ 2 3 1 ]
[ 0 -1 4 ]

is a 2 × 3 matrix because it has 2 rows and 3 columns. The entry in row (i) and column (j) is often denoted by (a_{ij}).

例如，矩阵

A = [ 2 3 1 ]
[ 0 -1 4 ]

是一个 2 × 3 矩阵，因为它有 2 行和 3 列。位于第 (i) 行第 (j) 列的元素通常记为 (a_{ij})。

A = [ a₁₁ a₁₂ a₁₃ ]
[ a₂₁ a₂₂ a₂₃ ]

In the example above, (a_{11} = 2), (a_{12} = 3), and (a_{23} = 4).

在上面的例子中，(a_{11} = 2)，(a_{12} = 3)，以及 (a_{23} = 4)。

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2. Addition and Subtraction of Matrices | 矩阵的加法与减法

Two matrices can be added or subtracted only if they have the same dimensions. We add or subtract corresponding entries.

只有当两个矩阵具有相同的维度时才能进行加法或减法运算。我们是对应元素相加或相减。

For example:

例如：

A = [ 1 2 ]
[ 3 4 ]  and  B = [ 5 6 ]
[ 7 8 ]

Then

那么

A + B = [ 6 8 ]
[ 10 12 ]

and

并且

A – B = [ -4 -4 ]
[ -4 -4 ]

• Addition is commutative: (A + B = B + A).
• 加法满足交换律：(A + B = B + A)。
• Addition is associative: ((A + B) + C = A + (B + C)).
• 加法满足结合律：((A + B) + C = A + (B + C))。
• Subtraction is not commutative.
• 减法不满足交换律。

If the dimensions do not match, the operation is undefined.

如果维度不匹配，则该运算没有定义。

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3. Scalar Multiplication | 标量乘法

When we multiply a matrix by a scalar (a single number), we multiply every entry by that scalar.

当我们用一个标量（单独一个数）乘以矩阵时，我们将每个元素都乘以这个标量。

For example, if

例如，如果

A = [ 2 -1 ]
[ 0 3 ]

then

那么

2A = [ 4 -2 ]
[ 0 6 ]

This operation is used frequently when solving equations involving matrices, such as (2X = A), where (X = frac{1}{2}A).

该运算在求解涉及矩阵的方程时经常使用，例如 (2X = A)，其中 (X = frac{1}{2}A)。

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4. Matrix Multiplication | 矩阵乘法

Matrix multiplication is more complex than addition. To multiply two matrices, the number of columns of the first matrix must equal the number of rows of the second matrix.

矩阵乘法比加法更复杂。要将两个矩阵相乘，第一个矩阵的列数必须等于第二个矩阵的行数。

If (A) is an (m times n) matrix and (B) is an (n times p) matrix, then the product (AB) is an (m times p) matrix. The entry in row (i) and column (j) of (AB) is obtained by multiplying the elements of row (i) of (A) by the corresponding elements of column (j) of (B), and then summing the products.

如果 (A) 是 (m times n) 矩阵，(B) 是 (n times p) 矩阵，则乘积 (AB) 是一个 (m times p) 矩阵。(AB) 中第 (i) 行第 (j) 列的元素，是通过将 (A) 的第 (i) 行元素与 (B) 的第 (j) 列对应元素相乘，然后将乘积相加得到的。

For example, let

例如，设

A = [ 1 2 ]
[ 3 4 ]  and  B = [ 5 6 ]
[ 7 8 ]

Then

那么

AB = [ (1×5 + 2×7) (1×6 + 2×8) ]
[ (3×5 + 4×7) (3×6 + 4×8) ]

AB = [ 19 22 ]
[ 43 50 ]

• Matrix multiplication is not commutative: in general, (AB neq BA).
• 矩阵乘法不满足交换律：一般情况下，(AB neq BA)。
• Matrix multiplication is associative: ((AB)C = A(BC)).
• 矩阵乘法满足结合律：((AB)C = A(BC))。
• The identity matrix (denoted (I)) acts like the number 1: (AI = IA = A).
• 单位矩阵（记为 (I)）的作用类似于数字 1：(AI = IA = A)。

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5. The Identity Matrix and Zero Matrix | 单位矩阵与零矩阵

The 2 × 2 identity matrix is

2 × 2 单位矩阵是

I = [ 1 0 ]
[ 0 1 ]

and the 3 × 3 identity matrix is

而 3 × 3 单位矩阵是

I = [ 1 0 0 ]
[ 0 1 0 ]
[ 0 0 1 ]

Multiplying any matrix by the identity matrix leaves it unchanged. The zero matrix, denoted (O), has all entries equal to 0. It satisfies (A + O = A).

任何矩阵乘以单位矩阵都保持不变。零矩阵，记为 (O)，所有元素均为 0。它满足 (A + O = A)。

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6. Determinant of a 2 × 2 Matrix | 二阶行列式

For a 2 × 2 matrix

对于一个 2 × 2 矩阵

A = [ a b ]
[ c d ]

the determinant is defined as

其行列式定义为

det(A) = ad – bc

The determinant is often written as (|A|). It can be positive, negative, or zero.

行列式通常也写为 (|A|)。它可以是正数、负数或零。

For example, if

例如，如果

A = [ 2 3 ]
[ 1 4 ]

then

那么

det(A) = 2×4 – 3×1 = 8 – 3 = 5

If (det(A) = 0), the matrix is called singular. A singular matrix has no inverse.

如果 (det(A) = 0)，则该矩阵称为奇异矩阵。奇异矩阵没有逆矩阵。

If (det(A) neq 0), the matrix is called non-singular and it has an inverse.

如果 (det(A) neq 0)，则该矩阵称为非奇异矩阵，它存在逆矩阵。

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7. Determinant of a 3 × 3 Matrix | 三阶行列式

For a 3 × 3 matrix

对于一个 3 × 3 矩阵

A = [ a b c ]
[ d e f ]
[ g h i ]

the determinant is given by expanding along the first row:

其行列式通过沿第一行展开得到：

det(A) = a(ei – fh) – b(di – fg) + c(dh – eg)

Be careful with the alternating signs: +, -, +.

注意交替符号：+、-、+。

For example, let

例如，设

A = [ 1 2 3 ]
[ 0 1 4 ]
[ 5 6 0 ]

Then

那么

det(A) = 1 × (1×0 – 4×6) – 2 × (0×0 – 4×5) + 3 × (0×6 – 1×5)

det(A) = 1 × (-24) – 2 × (-20) + 3 × (-5)

det(A) = -24 + 40 – 15 = 1

Since (det(A) = 1 neq 0), the matrix is non-singular.

由于 (det(A) = 1 neq 0)，该矩阵是非奇异的。

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8. Inverse of a 2 × 2 Matrix | 二阶逆矩阵

For a 2 × 2 matrix

对于一个 2 × 2 矩阵

A = [ a b ]
[ c d ]

the inverse is

其逆矩阵为

A⁻¹ = (1 / det(A)) × [ d -b ]
[ -c a ]

provided (det(A) neq 0).

前提是 (det(A) neq 0)。

For example, if

例如，如果

A = [ 2 3 ]
[ 1 4 ]

then (det(A) = 5), and

那么 (det(A) = 5)，并且

A⁻¹ = (1/5) × [ 4 -3 ]
[ -1 2 ]

A⁻¹ = [ 4/5 -3/5 ]
[ -1/5 2/5 ]

Check: (A A^{-1} = I).

验证：(A A^{-1} = I)。

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

= (1/5) × [ 5 0 ] = [ 1 0 ] = I
[ 0 5 ] [ 0 1 ]

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9. Inverse of a 3 × 3 Matrix | 三阶逆矩阵

For a 3 × 3 matrix, finding the inverse is more involved. The method is:

对于 3 × 3 矩阵，求逆矩阵的过程更为复杂。方法如下：

1. Calculate (det(A)). If it is zero, the matrix is singular and has no inverse.
2. 计算 (det(A))。如果为零，则该矩阵是奇异的，没有逆矩阵。
3. Find the matrix of minors.
4. 求余子式矩阵。
5. Apply the checkerboard signs to get the cofactor matrix.
6. 应用棋盘符号得到伴随矩阵。
7. Transpose the cofactor matrix to get the adjugate matrix.
8. 将伴随矩阵转置得到伴随矩阵（adjugate）。
9. Multiply by (1/det(A)).
10. 乘以 (1/det(A))。

This is often summarised as:

这通常总结为：

A⁻¹ = (1 / det(A)) × adj(A)

Although the full calculation for a 3 × 3 inverse is lengthy, it can be performed using the cofactor expansion method. In the exam, always check your answer by multiplying (A) and (A^{-1}) to confirm you get (I).

虽然 3 × 3 逆矩阵的完整计算很冗长，但可以通过余子式展开法来完成。在考试中，务必通过将 (A) 与 (A^{-1}) 相乘来验证结果是否为 (I)。

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10. Solving Simultaneous Equations Using Matrices | 用矩阵求解联立方程组

Matrices provide a systematic way to solve systems of linear equations. Consider the system

矩阵为求解线性方程组提供了一种系统性的方法。考虑以下方程组

2x + 3y = 7
x + 4y = 5

This can be written as

这可以写成

[ 2 3 ] [ x ] = [ 7 ]
[ 1 4 ] [ y ] [ 5 ]

or (AX = B), where

或 (AX = B)，其中

A = [ 2 3 ]
[ 1 4 ], X = [ x ]
[ y ], B = [ 7 ]
[ 5 ]

To solve, multiply both sides by (A^{-1}):

求解时，两边同时乘以 (A^{-1})：

X = A⁻¹B

We already found (A^{-1}) above:

我们上面已经求出了 (A^{-1})：

A⁻¹ = (1/5) [ 4 -3 ]
[ -1 2 ]

Therefore

因此

X = (1/5) [ 4 -3 ] [ 7 ]
[ -1 2 ] [ 5 ]

= (1/5) [ 28 – 15 ]
[ -7 + 10 ]

= (1/5) [ 13 ]
[ 3 ]

So (x = 13/5) and (y = 3/5).

所以 (x = 13/5)，(y = 3/5)。

If (det(A) = 0), the equations may have no unique solution: they are either inconsistent or dependent.

如果 (det(A) = 0)，方程组可能没有唯一解：它们要么是不相容的，要么是相关的。

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11. Geometric Transformations Using Matrices | 矩阵的几何变换

A 2 × 2 matrix can represent a linear transformation of the plane. The columns of the matrix tell us where the basis vectors ((1,0)) and ((0,1)) map to.

一个 2 × 2 矩阵可以表示平面的线性变换。矩阵的列告诉我们基向量 ((1,0)) 和 ((0,1)) 映射到了哪里。

• Reflection in the x-axis: [ 1 0 ]
  [ 0 -1 ]
• 关于 x 轴的反射： [ 1 0 ]
  [ 0 -1 ]
• Reflection in the y-axis: [ -1 0 ]
  [ 0 1 ]
• 关于 y 轴的反射： [ -1 0 ]
  [ 0 1 ]
• Reflection in the line y = x: [ 0 1 ]
  [ 1 0 ]
• 关于直线 y = x 的反射： [ 0 1 ]
  [ 1 0 ]
• Rotation anticlockwise by 90°: [ 0 -1 ]
  [ 1 0 ]
• 逆时针旋转 90°： [ 0 -1 ]
  [ 1 0 ]
• Rotation anticlockwise by 180°: [ -1 0 ]
  [ 0 -1 ]
• 逆时针旋转 180°： [ -1 0 ]
  [ 0 -1 ]
• Enlargement by scale factor k: [ k 0 ]
  [ 0 k ]
• 比例因子为 k 的放大： [ k 0 ]
  [ 0 k ]

To transform a point ((x,y)), write it as a column vector and multiply by the matrix:

要变换点 ((x,y))，将其写成列向量并与矩阵相乘：

[ x’ ] = M [ x ]
[ y’ ] [ y ]

For example, a 90° anticlockwise rotation of ((2,3)) gives

例如，((2,3)) 逆时针旋转 90° 得到

[ 0 -1 ] [ 2 ] = [ -3 ]
[ 1 0 ] [ 3 ] [ 2 ]

The point ((2,3)) maps to ((-3,2)).

点 ((2,3)) 映射到 ((-3,2))。

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12. Determinants and Area Scale Factors | 行列式与面积比例因子

The absolute value of the determinant of a transformation matrix gives the area scale factor of the transformation. For a shape whose original area is (A), the area of its image under the matrix (M) is

变换矩阵行列式的绝对值给出了该变换的面积比例因子。对于原始面积为 (A) 的图形，在矩阵 (M) 作用下像的面积是

Area of image = |det(M)| × Original area

For example, the matrix

例如，矩阵

M = [ 2 0 ]
[ 0 3 ]

has determinant 6. So it scales areas by a factor of 6. If a triangle has area 4, its image has area 24.

其行列式为 6。因此它将面积缩放 6 倍。如果一个三角形的面积为 4，则其像的面积为 24。

The sign of the determinant also tells us about orientation: a negative determinant means the transformation reverses orientation (e.g., reflection).

行列式的符号还告诉我们方向信息：负行列式意味着变换反转了方向（例如反射）。

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