Matrix Transformations | 矩阵变换

📚 Matrix Transformations | 矩阵变换

Matrix transformations are one of the most visual and rewarding topics in IGCSE Mathematics. They combine the algebraic power of matrices with the geometric intuition of shapes moving on a coordinate plane. In this revision guide, we will cover the essential transformation matrices, how to combine transformations, and how to interpret the determinant — everything you need for exam success.

矩阵变换是 IGCSE 数学中最直观也最有趣的主题之一。它将矩阵的代数力量与图形在坐标平面上移动的几何直觉结合在一起。在本复习指南中，我们将涵盖必备的变换矩阵、如何组合变换，以及如何理解行列式的意义——助你在考试中取得理想成绩。

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1. What is a Matrix Transformation? | 什么是矩阵变换？

A matrix transformation is a rule that maps every point (x, y) of a shape to a new point (x’, y’) using matrix multiplication. If the transformation matrix is M, and the position vector of a point is v, then the image is given by:

矩阵变换是一种规则，它将图形上的每一个点 (x, y) 通过矩阵乘法映射到一个新的点 (x’, y’)。如果变换矩阵为 M，点的位置向量为 v，则像点的计算公式为：

v’ = M × v

In column vector form, a point (x, y) is written as a 2×1 column vector. The transformation matrix is a 2×2 matrix. The result is the coordinates of the image point.

在列向量形式中，点 (x, y) 写作一个 2×1 列向量。变换矩阵是一个 2×2 矩阵。计算结果就是像点的坐标。

The identity matrix I = [[1, 0], [0, 1]] leaves every point unchanged. It is the matrix equivalent of “doing nothing” and serves as the neutral element in matrix multiplication.

单位矩阵 I = [[1, 0], [0, 1]] 不改变任何点的位置。它是矩阵中“什么都不做”的变换，也是矩阵乘法中的中性元素。

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2. The Four Basic Transformations | 四种基本变换

There are four types of transformations you must know: reflection, rotation, enlargement, and translation. However, in IGCSE matrix work, the first three are represented by 2×2 matrices; translation is handled separately by adding a vector.

你必须掌握四种基本变换：反射、旋转、放缩（扩大/缩小）和平移。但在 IGCSE 矩阵部分中，前三种用 2×2 矩阵表示；平移则通过加一个向量单独处理。

• Reflection | 反射：A shape is flipped over a mirror line. The image is congruent to the original.

  反射：图形关于某条镜面线翻转。像与原图形全等。

• Rotation | 旋转：A shape is turned about a fixed point through a given angle. The image is congruent.

  旋转：图形绕一个固定点旋转给定的角度。像与原图形全等。

• Enlargement | 放缩：A shape is resized by a scale factor, with a centre of enlargement. The image is similar.

  放缩：图形以某个放大中心按比例因子改变大小。像与原图形相似。

• Translation | 平移：A shape slides by a vector. The image is congruent.

  平移：图形沿某个向量滑动。像与原图形全等。

For matrices, you only need the first three. Translation is done by adding a column vector to the coordinates, not by a 2×2 matrix.

对于矩阵部分，你只需要掌握前三种。平移通过向坐标加上一个列向量来实现，而不是通过 2×2 矩阵。

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3. Reflection Matrices You Must Memorise | 必须牢记的反射矩阵

Reflection matrices depend entirely on the mirror line. Here are the ones that appear most frequently in IGCSE exams.

反射矩阵完全取决于镜面线的位置。下面是 IGCSE 考试中最常出现的几个。

Mirror Line | 镜面线	Matrix | 矩阵
x-axis (y = 0) | x 轴	[[1, 0], [0, −1]]
y-axis (x = 0) | y 轴	[[−1, 0], [0, 1]]
y = x | 直线 y = x	[[0, 1], [1, 0]]
y = −x | 直线 y = −x	[[0, −1], [−1, 0]]

Notice the pattern: reflecting in the x-axis negates the y-coordinate only; reflecting in y = x swaps x and y. You can verify these by applying the matrix to the unit point (1, 0).

注意规律：关于 x 轴反射只改变 y 坐标的符号；关于 y = x 反射则交换 x 和 y。你可以通过将矩阵作用于单位点 (1, 0) 来验证。

For any general straight line through the origin, a reflection matrix can be derived using trigonometry, but for IGCSE, you are usually only asked to use the four special cases above, or to work out the matrix for a reflection described in a question.

对于任何通过原点的直线，反射矩阵可以通过三角函数推导得出，但 IGCSE 通常只要求你使用以上四种特殊情况，或者根据题目描述推导矩阵。

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4. Rotation Matrices | 旋转矩阵

In IGCSE, rotations are taken about the origin (0, 0). The angle is measured anticlockwise as the positive direction.

在 IGCSE 中，旋转都是绕原点 (0, 0) 进行的。逆时针方向为角度的正方向。

Rotation | 旋转	Matrix | 矩阵
90° anticlockwise | 逆时针 90°	[[0, −1], [1, 0]]
180° | 旋转 180°	[[−1, 0], [0, −1]]
270° anticlockwise (or 90° clockwise) | 逆时针 270°（即顺时针 90°）	[[0, 1], [−1, 0]]

The general rotation matrix through an angle θ anticlockwise about the origin is:

绕原点逆时针旋转 θ 角的一般旋转矩阵为：

[[cos θ, −sin θ], [sin θ, cos θ]]

If the question states a clockwise rotation, simply use a negative angle. For example, a 90° clockwise rotation is the same as a 270° anticlockwise rotation, giving the matrix [[0, 1], [−1, 0]].

如果题目要求顺时针旋转，只需使用负角度。例如，顺时针旋转 90° 等同于逆时针旋转 270°，其矩阵为 [[0, 1], [−1, 0]]。

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5. Enlargement | 放缩变换

An enlargement with scale factor k and centre at the origin is represented by the matrix:

以原点为中心、比例因子为 k 的放缩变换矩阵为：

[[k, 0], [0, k]]

This multiplies both coordinates by k. If k > 1, the shape gets larger; if 0 < k < 1, the shape gets smaller; if k is negative, the shape is enlarged (or reduced) and rotated 180°.

该矩阵将两个坐标同时乘以 k。如果 k > 1，图形变大；如果 0 < k < 1，图形变小；如果 k 为负数，则图形在放大（或缩小）的同时旋转 180°。

Enlargements with a centre other than the origin are not represented by a simple 2×2 matrix in the IGCSE syllabus — those require translations before and after scaling. In matrix transformation questions, the centre is always the origin unless stated otherwise.

中心不在原点的放缩在 IGCSE 考纲中不通过简单的 2×2 矩阵表示——这种变换需要在放缩前后进行平移。在矩阵变换题目中，除非特别说明，中心始终是原点。

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6. Combining Transformations | 组合变换

When two or more transformations are applied in sequence, the combined transformation is the product of their matrices — but the order matters!

当依次应用两个或更多变换时，组合变换等于它们矩阵的乘积——但顺序很重要！

Important rule: If transformation A is applied first, followed by transformation B, then the combined matrix is:

重要规则：如果先进行变换 A，再进行变换 B，则组合矩阵为：

M_combined = B × A

In other words, the transformation applied first goes on the right. This is because the position vector is multiplied on the right: first by A, then by B. Matrices are not commutative — BA is generally not equal to AB.

换句话说，先进行的变换矩阵放在右边。这是因为位置向量从右侧乘起：先乘 A，再乘 B。矩阵乘法不满足交换律——BA 通常不等于 AB。

Let us look at a worked example. Suppose a shape is reflected in the y-axis (matrix A = [[−1, 0], [0, 1]]) and then rotated 90° anticlockwise (matrix B = [[0, −1], [1, 0]]). The combined matrix is:

我们来看一个例子。假设一个图形先关于 y 轴反射（矩阵 A = [[−1, 0], [0, 1]]），再逆时针旋转 90°（矩阵 B = [[0, −1], [1, 0]]）。组合矩阵为：

M = B × A = [[0, −1], [1, 0]] × [[−1, 0], [0, 1]] = [[0, −1], [−1, 0]]

The result is a reflection in the line y = −x. If you reverse the order, you would get a different matrix, so always check which transformation comes first.

结果是关于直线 y = −x 的反射。如果颠倒顺序，你将得到不同的矩阵，所以务必先确认哪个变换在前。

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7. The Determinant and Area | 行列式与面积

The determinant of a 2×2 matrix [[a, b], [c, d]] is defined as:

2×2 矩阵 [[a, b], [c, d]] 的行列式定义为：

det = ad − bc

The absolute value of the determinant gives the area scale factor of the transformation. If a shape has area A, its image after the transformation has area |det| × A.

行列式的绝对值给出了变换的面积比例因子。如果一个图形的面积为 A，则其变换后像的面积为 |det| × A。

This is a powerful tool for solving area problems quickly. For example, if a triangle of area 5 cm² is transformed by a matrix with determinant 3, the image triangle has area 15 cm².

这是一个快速解决面积问题的有力工具。例如，若一个面积为 5 cm² 的三角形经行列式为 3 的矩阵变换，则像三角形的面积为 15 cm²。

If the determinant is 0, the transformation collapses the shape into a line or a point — the shape loses all its area. This is called a singular transformation.

如果行列式为 0，变换将图形压缩为一条线或一个点——图形失去了所有面积。这称为奇异变换。

Negative determinants indicate that the orientation of the shape has been reversed — for example, a reflection has determinant −1 for any mirror line through the origin.

行列式为负表示图形的方向被反转——例如，关于任何过原点的直线反射，其行列式均为 −1。

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8. Inverse Transformations | 逆变换

If a transformation matrix M maps a shape to its image, the inverse matrix M⁻¹ maps the image back to the original shape. For a 2×2 matrix:

如果变换矩阵 M 将图形映射到其像，则逆矩阵 M⁻¹ 将像映射回原图形。对于 2×2 矩阵：

M⁻¹ = 1/(ad − bc) × [[d, −b], [−c, a]]

The inverse exists only when ad − bc ≠ 0. If the determinant is zero, the transformation cannot be reversed because information about the original shape has been lost.

只有当 ad − bc ≠ 0 时逆矩阵才存在。如果行列式为 0，则变换不可逆，因为原图形的信息已经丢失。

Geometrically, the inverse of a reflection is the same reflection; the inverse of a 90° anticlockwise rotation is a 90° clockwise rotation (i.e., a 270° anticlockwise rotation); the inverse of an enlargement by factor k is an enlargement by factor 1/k.

从几何上看，反射的逆变换是其自身；逆时针旋转 90° 的逆变换是顺时针旋转 90°（即逆时针旋转 270°）；比例因子为 k 的放缩的逆变换是比例因子为 1/k 的放缩。

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9. Worked Example | 例题精讲

Let us apply everything together. A quadrilateral has vertices at A(1, 0), B(2, 1), C(1, 3), D(0, 1). It is transformed by the matrix M = [[2, 0], [0, 2]], followed by a reflection in the x-axis.

让我们综合运用所学的内容。一个四边形顶点为 A(1, 0), B(2, 1), C(1, 3), D(0, 1)。它先经过矩阵 M = [[2, 0], [0, 2]] 变换，再进行关于 x 轴的反射。

Step 1: Identify the two matrices. The first is an enlargement by factor 2. The second is the reflection matrix R = [[1, 0], [0, −1]]. Since the enlargement is applied first, the combined matrix is:

第一步：确定两个矩阵。第一个是比例因子为 2 的放缩。第二个是反射矩阵 R = [[1, 0], [0, −1]]。由于放缩先进行，组合矩阵为：

M_total = R × M = [[1, 0], [0, −1]] × [[2, 0], [0, 2]] = [[2, 0], [0, −2]]

Step 2: Apply M_total to each vertex. For example, A(1, 0):

第二步：将 M_total 作用于每个顶点。例如，A(1, 0)：

[[2, 0], [0, −2]] × [[1],[0]] = [[2],[0]]

So A maps to A'(2, 0). Similarly, B(2, 1) maps to (4, −2); C(1, 3) maps to (2, −6); D(0, 1) maps to (0, −2).

所以 A 映射到 A'(2, 0)。同样地，B(2, 1) 映射到 (4, −2)；C(1, 3) 映射到 (2, −6)；D(0, 1) 映射到 (0, −2)。

Step 3: The area of the original quadrilateral can be found using the shoelace formula or by splitting into triangles. The determinant of M_total is 2 × (−2) = −4, so its absolute value is 4. The image area is 4 times the original area.

第三步：原四边形的面积可以用鞋带公式或分割成三角形来求得。M_total 的行列式为 2 × (−2) = −4，其绝对值为 4。像的面积是原面积的 4 倍。

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10. Exam Tips and Common Mistakes | 考试技巧与常见错误

Matrix transformation questions are a reliable source of marks if you are careful. Here are the most important tips to avoid losing marks.

矩阵变换题目只要你够细心，是稳定的得分点。以下是最重要的提分和避错技巧。

• Order of multiplication: Always write the first transformation on the right when combining. Check the wording of the question carefully.

  乘法顺序：组合变换时，始终将先进行的变换写在右侧。仔细阅读题目措辞。

• Direction of rotation: The positive direction is anticlockwise. A clockwise 90° rotation is not the same matrix as an anticlockwise 90° rotation.

  旋转方向：正方向为逆时针。顺时针旋转 90° 与逆时针旋转 90° 的矩阵不同。

• Use unit points: To verify a matrix, apply it to (1, 0) and (0, 1) and check whether the images match the expected transformation.

  利用单位点：要验证矩阵是否正确，将其作用于 (1, 0) 和 (0, 1)，检查结果是否符合预期变换。

• Area scale factor: The area scale factor is |det|, not det. A negative determinant still gives a positive area scale factor.

  面积比例因子：面积比例因子是 |det|，不是 det。行列式为负时，面积比例因子仍然为正。

• Column vector form: Always write points as column vectors when multiplying by a matrix. Writing them as row vectors will give incorrect results.

  列向量形式：与矩阵相乘时始终将点写成列向量。如果写成行向量，结果将是错误的。

Practice drawing the shape and its image on graph paper for every question. Visual verification is fast and catches many algebraic mistakes.

在方格纸上画出原图形和像图形进行练习。视觉验证速度很快，而且能捕捉到许多代数错误。

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11. Quick Reference Table | 快速参考表

Here is a summary of all the matrices you should know for the exam — keep this table in mind as a final revision checklist.

以下是考试中你应该掌握的所有矩阵汇总——请将此表作为最后复习清单。

Transformation | 变换	Matrix | 矩阵	Determinant | 行列式
Reflection in x-axis | x 轴反射	[[1, 0], [0, −1]]	−1
Reflection in y-axis | y 轴反射	[[−1, 0], [0, 1]]	−1
Reflection in y = x | y = x 反射	[[0, 1], [1, 0]]	−1
Reflection in y = −x | y = −x 反射	[[0, −1], [−1, 0]]	−1
Rotation 90° anticlockwise | 逆时针旋转 90°	[[0, −1], [1, 0]]	1
Rotation 180° | 旋转 180°	[[−1, 0], [0, −1]]	1
Rotation 270° anticlockwise | 逆时针旋转 270°	[[0, 1], [−1, 0]]	1
Enlargement by factor k | 比例因子 k 的放缩	[[k, 0], [0, k]]	k²

Memorising these eight matrices will cover the overwhelming majority of IGCSE matrix transformation questions. Combine them, compute their determinants, and you are equipped to handle anything the exam paper throws at you.

牢记这八个矩阵足以应对绝大多数 IGCSE 矩阵变换题目。组合它们，计算行列式，你就能从容应对试卷上的任何问题。

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