Determinants | 行列式

📚 Determinants | 行列式

In A-Level mathematics, the determinant is a scalar value associated with a square matrix. It reveals crucial information about the matrix: whether it is invertible, how it transforms areas or volumes, and whether a system of linear equations has a unique solution. This article covers the full AQA specification on determinants, from 2×2 and 3×3 calculations to geometric interpretations and exam-style applications.

在 A-Level 数学中，行列式是与方阵相关联的一个标量值。它揭示了关于矩阵的关键信息：矩阵是否可逆、如何变换面积或体积，以及线性方程组是否具有唯一解。本文完整覆盖 AQA 考纲中关于行列式的内容，从 2×2 与 3×3 的计算到几何意义和考试型应用。

1. What is a Determinant? | 什么是行列式？

A determinant is a number computed from the entries of a square matrix. For a 2×2 matrix, it is the difference between the products of the two diagonals. For a 3×3 matrix, the computation is more involved but follows a systematic pattern. The determinant is written using vertical bars: the determinant of matrix A is denoted det(A) or |A|.

行列式是由方阵中的元素计算得出的一个数值。对于 2×2 矩阵，它是两条对角线乘积之差。对于 3×3 矩阵，计算过程较为复杂，但遵循系统性的规律。行列式用竖线表示：矩阵 A 的行列式记为 det(A) 或 |A|。

Only square matrices have determinants. If a matrix is not square (the number of rows does not equal the number of columns), its determinant is undefined. In the AQA specification, you are required to evaluate determinants of 2×2 and 3×3 matrices, apply their properties, and link them to inverse matrices and geometric transformations.

只有方阵才有行列式。如果矩阵不是方阵（行数不等于列数），则其行列式无定义。在 AQA 考纲中，你被要求计算 2×2 和 3×3 矩阵的行列式，运用它们的性质，并将其与逆矩阵和几何变换联系起来。

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

For a 2×2 matrix, the formula is straightforward. Given matrix A = [a b; c d] (in row notation), the determinant is calculated as ad − bc. It is the product of the main diagonal entries minus the product of the secondary diagonal entries.

对于 2×2 矩阵，公式非常直接。给定矩阵 A = [a b; c d]（按行表示），行列式计算为 ad − bc。即主对角线元素乘积减去副对角线元素乘积。

det(A) = ad − bc

For example, if A = [3 4; 2 5], then det(A) = 3 × 5 − 4 × 2 = 15 − 8 = 7. A positive determinant indicates that the orientation of space is preserved (a rotation or scaling), while a negative determinant indicates a reflection has occurred.

例如，若 A = [3 4; 2 5]，则 det(A) = 3 × 5 − 4 × 2 = 15 − 8 = 7。正行列式表明空间方向被保持（旋转或缩放），而负行列式表明发生了反射。

If det(A) = 0, the matrix is called singular (non-invertible).
If det(A) ≠ 0, the matrix is non-singular and has an inverse.

若 det(A) = 0，该矩阵称为奇异矩阵（不可逆）。
若 det(A) ≠ 0，该矩阵为非奇异矩阵，存在逆矩阵。

It is essential to substitute the correct entries into the formula. A common exam error is to use ac − bd instead of ad − bc. Always label the matrix entries systematically before computing.

将正确的元素代入公式至关重要。一个常见的考试错误是误用 ac − bd 而不是 ad − bc。在计算之前，务必系统地标注矩阵元素。

3. Determinant of a 3×3 Matrix | 3×3 矩阵的行列式

For a 3×3 matrix, we use a method called expansion along a row or column. The standard approach in AQA is to expand along the first row, using a pattern of alternating signs. Given a 3×3 matrix:

对于 3×3 矩阵，我们使用按行或按列展开的方法。AQA 中的标准做法是沿第一行展开，采用交替正负号的模式。给定一个 3×3 矩阵：

|a b c; d e f; g h i| = a(ei − fh) − b(di − fg) + c(dh − eg)

Notice the sign pattern: +, −, + for the first row. Each 2×2 determinant is formed by deleting the row and column containing the corresponding multiplier. For the term a, we delete the first row and first column, leaving the 2×2 matrix [e f; h i]. For the term b, we delete the first row and second column, leaving [d f; g i]. For the term c, we delete the first row and third column, leaving [d e; g h].

注意符号模式：第一行依次为 +、−、+。每个 2×2 行列式是通过删去对应乘数所在的行和列形成的。对于 a 项，删去第一行第一列，剩下 2×2 矩阵 [e f; h i]。对于 b 项，删去第一行第二列，剩下 [d f; g i]。对于 c 项，删去第一行第三列，剩下 [d e; g h]。

For example, calculate the determinant of A = [1 2 3; 0 1 4; 5 6 0]. Using the formula:

例如，计算 A = [1 2 3; 0 1 4; 5 6 0] 的行列式。使用公式：

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

= 1(0 − 24) − 2(0 − 20) + 3(0 − 5) = −24 + 40 − 15 = 1

Although expanding along the first row is standard, you may expand along any row or column. Expanding along a row or column with zeros can simplify calculations significantly. The sign pattern for a 3×3 matrix is:

虽然按第一行展开是标准方法，但你可以沿任意行或列展开。沿含有零的行或列展开可以显著简化计算。3×3 矩阵的符号模式为：

+ − +; − + −; + − +

4. Geometric Meaning of the Determinant | 行列式的几何意义

For a 2×2 matrix, the absolute value of the determinant represents the area scale factor of the transformation. If a unit square is transformed by the matrix, the area of the resulting parallelogram equals |det(A)|. The sign of the determinant indicates whether the orientation of the shape is reversed.

对于 2×2 矩阵，行列式的绝对值表示变换的面积缩放因子。如果一个单位正方形被该矩阵变换，所得平行四边形的面积等于 |det(A)|。行列式的符号指示形状的方向是否被反转。

For a 3×3 matrix, the absolute value of the determinant represents the volume scale factor. A unit cube transformed by the matrix has volume |det(A)|. This geometric interpretation is often examined in questions that involve transformations of the plane or space.

对于 3×3 矩阵，行列式的绝对值表示体积缩放因子。单位立方体被该矩阵变换后，其体积为 |det(A)|。这种几何解释常出现在涉及平面或空间变换的考题中。

The area of a triangle with vertices at the origin, (x₁, y₁), and (x₂, y₂) is given by half the absolute value of the 2×2 determinant:

顶点在原点、(x₁, y₁) 和 (x₂, y₂) 的三角形的面积等于 2×2 行列式绝对值的一半：

Area = ½|x₁y₂ − x₂y₁|

Similarly, the area of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) can be found using a 3×3 determinant with a row of ones:

类似地，顶点为 (x₁, y₁)、(x₂, y₂) 和 (x₃, y₃) 的三角形面积可以用带有一行 1 的 3×3 行列式求得：

Area = ½ |det[x₁ y₁ 1; x₂ y₂ 1; x₃ y₃ 1]|

5. Determinant and the Inverse Matrix | 行列式与逆矩阵

The determinant plays a central role in finding the inverse of a matrix. For a 2×2 matrix A = [a b; c d], if det(A) ≠ 0, the inverse is given by:

行列式在求矩阵逆矩阵方面发挥着核心作用。对于 2×2 矩阵 A = [a b; c d]，若 det(A) ≠ 0，则逆矩阵为：

A⁻¹ = (1/det(A)) × [d −b; −c a]

The 1/det(A) factor is a scalar multiplier. Since division by zero is undefined, a zero determinant means no inverse exists. This is a direct link: a matrix is invertible if and only if its determinant is non-zero.

1/det(A) 是一个标量乘数。由于除以零无定义，行列式为零意味着不存在逆矩阵。这构成直接联系：矩阵可逆当且仅当其行列式非零。

For a 3×3 matrix, the inverse is computed using the adjugate matrix: A⁻¹ = (1/det(A)) × adj(A). Although the full computation of the adjugate is beyond the scope of some exam questions, you must understand that the inverse exists only when det(A) ≠ 0, and that calculating the determinant is always the first step.

对于 3×3 矩阵，逆矩阵通过伴随矩阵计算：A⁻¹ = (1/det(A)) × adj(A)。虽然伴随矩阵的完整计算超出了某些考题的范围，但你必须理解逆矩阵仅在 det(A) ≠ 0 时存在，并且计算行列式始终是第一步。

In solving matrix equations, if AX = B and A is invertible, then X = A⁻¹B. The determinant assures us the solution is unique. If det(A) = 0, the system is either inconsistent or has infinitely many solutions.

在解矩阵方程时，若 AX = B 且 A 可逆，则 X = A⁻¹B。行列式确保解是唯一的。若 det(A) = 0，则方程组要么无解，要么有无穷多解。

6. Properties of Determinants | 行列式的性质

Several properties of determinants are frequently tested in exams. Knowing these can save time and help you check answers. Here are the most important ones:

行列式的几个性质在考试中经常被考查。了解这些性质可以节省时间并帮助你检查答案。以下是最重要的性质：

If a matrix has a row or column of zeros, its determinant is zero.
Swapping two rows (or two columns) multiplies the determinant by −1.
Multiplying a single row by a scalar k multiplies the determinant by k.
Multiplying the entire matrix by a scalar k multiplies the determinant by kⁿ for an n×n matrix.
The determinant of a matrix equals the determinant of its transpose: det(A) = det(Aᵀ).
For two square matrices A and B of the same size, det(AB) = det(A) × det(B).
If A is invertible, det(A⁻¹) = 1/det(A).

若矩阵的某行或某列全为零，则其行列式为零。
交换两行（或两列）会使得行列式乘以 −1。
将某一行乘以标量 k 会使行列式乘以 k。
将整个矩阵乘以标量 k 会使 n×n 矩阵的行列式乘以 kⁿ。
矩阵的行列式等于其转置矩阵的行列式：det(A) = det(Aᵀ)。
对于两个同阶方阵 A 和 B，det(AB) = det(A) × det(B)。
若 A 可逆，则 det(A⁻¹) = 1/det(A)。

Another important result: a matrix with two identical rows or two proportional rows has a determinant of zero. This follows from the row-swap property, since swapping them does not change the matrix but must change the sign of the determinant, implying the determinant is zero.

另一个重要结果：如果矩阵有两行相同或两行成比例，则其行列式为零。这可以由此前的行交换性质推出，因为交换它们不会改变矩阵本身，但必须改变行列式的符号，从而推出行列式为零。

7. Solving Equations Using Determinants | 用行列式解方程组

For a system of linear equations written in matrix form AX = B, where A is a 2×2 or 3×3 coefficient matrix, the determinant is used to determine solvability. If det(A) ≠ 0, the system has a unique solution given by X = A⁻¹B. If det(A) = 0, the system either has no solution or infinitely many solutions.

对于写成矩阵形式 AX = B 的线性方程组，其中 A 是 2×2 或 3×3 系数矩阵，行列式用于判断可解性。若 det(A) ≠ 0，方程组有唯一解，即 X = A⁻¹B。若 det(A) = 0，则方程组无解或有无穷多解。

Cramer’s rule provides an alternative method for solving systems using determinants. For a 2×2 system a₁x + b₁y = c₁; a₂x + b₂y = c₂, the solution is:

克莱默法则提供了使用行列式解方程组的另一种方法。对于 2×2 方程组 a₁x + b₁y = c₁; a₂x + b₂y = c₂，解为：

x = det[c₁ b₁; c₂ b₂] / det[a₁ b₁; a₂ b₂]

y = det[a₁ c₁; a₂ c₂] / det[a₁ b₁; a₂ b₂]

This method is elegant because it reduces solving to determinant evaluation. However, be cautious: if the denominator determinant equals zero, Cramer’s rule cannot be applied, and you must analyse the system separately.

这种方法很优雅，因为它将求解简化为行列式求值。但要谨慎：如果分母行列式为零，则不能使用克莱默法则，必须单独分析方程组。

In a 3×3 system, for a unique solution to exist, the determinant of the coefficient matrix must be non-zero. This is a reliable test before attempting any computation-heavy solving method.

在 3×3 方程组中，要存在唯一解，系数矩阵的行列式必须非零。在尝试任何计算量大的求解方法之前，这是一个可靠的检验手段。

8. Exam Tips and Common Errors | 考试技巧与常见错误

In AQA exam papers, determinant questions often appear as part of a larger problem on matrices and transformations. Here are the most frequent pitfalls and how to avoid them:

在 AQA 试卷中，行列式问题通常作为矩阵和变换综合题的一部分出现。以下是最常见的陷阱以及如何避免它们：

Sign errors in 3×3 expansion: Remember the alternating signs +, −, + for the first row. Misplacing signs is the leading cause of wrong answers.
Mismatched diagonal products: For 2×2 matrices, always compute ad − bc, not ab − cd or ac − bd.
Scalar multiplication of a matrix vs. row: Multiplying a whole 2×2 matrix by k multiplies the determinant by k², not k.
Zero determinant conclusion: det(A) = 0 means the matrix is singular and cannot be inverted, but it does not mean the transformation collapses everything to a point — it means area or volume factors to zero.

3×3 展开中的符号错误：记住第一行的交替符号 +、−、+。符号放错是答错的首要原因。
对角线乘积配错：对于 2×2 矩阵，务必计算 ad − bc，而不是 ab − cd 或 ac − bd。
矩阵乘标量与行乘标量混淆：将整个 2×2 矩阵乘以 k 会使行列式乘以 k²，而不是 k。
零行列式的结论：det(A) = 0 意味着矩阵是奇异的且不可逆，但这并不意味着变换将所有东西压缩到一点——只是面积或体积缩放为零。

To verify a computed determinant, substitute the matrix into the area formula or use the property det(AB) = det(A)det(B) to cross-check with a known transformation. Quick verification is an excellent habit in timed exam conditions.

要验证计算出的行列式，可以将矩阵代入面积公式，或利用性质 det(AB) = det(A)det(B) 与已知变换交叉验证。在限时考试条件下，快速验证是一个非常好的习惯。

9. Worked Example: Applying Multiple Concepts | 综合例题：多个概念的应用

Consider the matrix A = [2 1; 4 3] and B = [3 2; 1 5]. First, compute det(A) = 2×3 − 1×4 = 6 − 4 = 2. Next, compute det(B) = 3×5 − 2×1 = 15 − 2 = 13. Now compute the product AB = [2×3+1×1, 2×2+1×5; 4×3+3×1, 4×2+3×5] = [7, 9; 15, 23]. Hence det(AB) = 7×23 − 9×15 = 161 − 135 = 26.

考虑矩阵 A = [2 1; 4 3] 和 B = [3 2; 1 5]。首先，计算 det(A) = 2×3 − 1×4 = 6 − 4 = 2。接着，计算 det(B) = 3×5 − 2×1 = 15 − 2 = 13。然后计算乘积 AB = [2×3+1×1, 2×2+1×5; 4×3+3×1, 4×2+3×5] = [7, 9; 15, 23]。因此 det(AB) = 7×23 − 9×15 = 161 − 135 = 26。

Notice that det(AB) = 26 = det(A) × det(B) = 2 × 13. This confirms the multiplicative property and gives you confidence that both determinants were computed correctly. This verification technique is particularly useful in multi-step exam questions.

注意 det(AB) = 26 = det(A) × det(B) = 2 × 13。这验证了乘积性质，并使你有信心认为两个行列式都计算正确。这种验证技巧在多步骤考试题中特别有用。

Now check invertibility. Since det(A) = 2 ≠ 0, A has an inverse, A⁻¹ = ½[3 −1; −4 2] = [1.5 −0.5; −2 1]. The determinant of A⁻¹ is 1.5×1 − (−0.5)(−2) = 1.5 − 1 = 0.5, which equals 1/det(A) = ½.

现在检查可逆性。由于 det(A) = 2 ≠ 0，A 存在逆矩阵，A⁻¹ = ½[3 −1; −4 2] = [1.5 −0.5; −2 1]。A⁻¹ 的行列式为 1.5×1 − (−0.5)(−2) = 1.5 − 1 = 0.5，等于 1/det(A) = ½。

10. Summary and Revision Checklist | 总结与复习清单

The determinant is a fundamental concept linking matrices, transformations, and linear equations. For a 2×2 matrix, remember ad − bc. For a 3×3 matrix, expand carefully with alternating signs. Geometrically, the determinant is the area or volume scale factor, and it indicates orientation through its sign.

行列式是一个基本概念，将矩阵、变换和线性方程组联系起来。对于 2×2 矩阵，记住 ad − bc。对于 3×3 矩阵，用交替符号仔细展开。在几何上，行列式是面积或体积缩放因子，并通过其符号指示方向。

Be able to compute det for 2×2 and 3×3 matrices without errors.
Understand the link between det = 0 and singularity/non-invertibility.
Use the inverse formula for 2×2 with the 1/det factor.
Apply geometric interpretations to area and volume problems.
Verify solutions using det(AB) = det(A)det(B).
In 3×3 expansion, expand along rows/columns with zeros when possible.

能够准确无误地计算 2×2 和 3×3 矩阵的行列式。
理解 det = 0 与奇异/不可逆之间的联系。
掌握带 1/det 因子的 2×2 逆矩阵公式。
将几何解释应用于面积和体积问题。
使用 det(AB) = det(A)det(B) 验证解。
在 3×3 展开时，尽可能沿含零的行或列展开。

Mastering determinants is not just about memorising formulas; it is about understanding the structure of linear algebra. Once you can confidently compute and interpret determinants, topics like inverse matrices, transformations, and systems of equations become significantly easier. Practice past paper questions and always check the sign pattern in 3×3 expansion. These steps will ensure you earn full marks on determinant questions.

掌握行列式不仅仅是记住公式，更是理解线性代数的结构。一旦你能自信地计算和解释行列式，逆矩阵、变换和方程组等主题就会变得容易得多。练习历年真题，并始终检查 3×3 展开中的符号模式。这些步骤将确保你在行列式题目上获得满分。

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Determinants | 行列式