Determinants | 行列式

📚 Determinants | 行列式

Determinants are a fundamental tool in linear algebra, commonly introduced in A-Level Mathematics. They encode essential information about a square matrix, such as whether it is invertible, and they provide a direct method for solving systems of linear equations and calculating areas and volumes.

行列式是线性代数中的基本工具，在 A-Level 数学中通常会正式引入。它编码了方阵的关键信息，例如矩阵是否可逆，并为求解线性方程组、计算面积与体积提供了直接的方法。

1. Definition of a Determinant | 行列式的定义

A determinant is a scalar value that can be computed from the elements of a square matrix. It is denoted by det(A), |A|, or by replacing the brackets of the matrix with vertical bars.

行列式是一个能从方阵元素计算得到的标量值。它记作 det(A)、|A|，或者将矩阵的方括号替换为竖线来表示。

For a 2 × 2 matrix A = [ a b ; c d ], the determinant is defined as:

det(A) = ad − bc

For a 3 × 3 matrix, the determinant can be found by expanding along a row or column using cofactors, which will be explained in a later section.

对于 3 × 3 矩阵，行列式可以通过按某一行或某一列进行余子式展开来计算，这一点将在后面的章节中说明。

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

Given a 2 × 2 matrix A = [ a b ; c d ], the determinant is the product of the main diagonal entries minus the product of the off-diagonal entries.

给定二阶矩阵 A = [ a b ; c d ]，行列式等于主对角线元素乘积减去副对角线元素乘积。

|A| = ad − bc

Example: For A = [ 3 4 ; 2 5 ], |A| = 3 × 5 − 4 × 2 = 15 − 8 = 7.

示例：对于 A = [ 3 4 ; 2 5 ]，|A| = 3 × 5 − 4 × 2 = 15 − 8 = 7。

If |A| = 0, the matrix is called singular, meaning it has no inverse. If |A| ≠ 0, the matrix is non-singular and an inverse exists.

若 |A| = 0，矩阵称为奇异矩阵，意味着它没有逆矩阵。若 |A| ≠ 0，则矩阵非奇异，逆矩阵存在。

3. Determinant of a 3 × 3 Matrix | 三阶矩阵的行列式

For a 3 × 3 matrix A = [ a b c ; d e f ; g h i ], the determinant can be computed by expanding along the first row:

对于三阶矩阵 A = [ a b c ; d e f ; g h i ]，行列式可按第一行展开计算：

|A| = a(ei − fh) − b(di − fg) + c(dh − eg)

Notice the alternating signs: +, −, +. The minor of an element is the determinant of the 2 × 2 matrix left after removing the row and column containing that element.

注意符号交替：+、−、+。一个元素的余子式是删去该元素所在行和列后剩下的 2 × 2 矩阵的行列式。

Example: For A = [ 1 2 3 ; 4 5 6 ; 7 8 9 ],

示例：对于 A = [ 1 2 3 ; 4 5 6 ; 7 8 9 ]，

|A| = 1(5·9 − 6·8) − 2(4·9 − 6·7) + 3(4·8 − 5·7) = 1(45 − 48) − 2(36 − 42) + 3(32 − 35) = −3 + 12 − 9 = 0

This matrix is singular because its third row is the sum of the first two rows? Actually, row3 − row2 = row2 − row1, so rows are linearly dependent.

该矩阵是奇异的，因为它的行向量线性相关。

4. Cofactor Expansion (Laplace Expansion) | 余子式展开（拉普拉斯展开）

The determinant can be expanded along any row or column. For element aᵢⱼ, its cofactor Cᵢⱼ is defined as Cᵢⱼ = (−1)^(i+j) × Mᵢⱼ, where Mᵢⱼ is the minor.

行列式可以沿任意一行或一列展开。对于元素 aᵢⱼ，其余子式 Cᵢⱼ 定义为 Cᵢⱼ = (−1)^(i+j) × Mᵢⱼ，其中 Mᵢⱼ 是该元素的余子式。

The sign pattern for a 3 × 3 matrix is:

3 × 3 矩阵的符号模式为：

+	−	+
−	+	−
+	−	+

Expanding along row 2 gives |A| = −d(bi − ch) + e(ai − cg) − f(ah − bg).

沿第二行展开得到 |A| = −d(bi − ch) + e(ai − cg) − f(ah − bg)。

Choosing a row or column with many zeros reduces the amount of computation significantly.

选择含零较多的行或列可以显著减少计算量。

5. Properties of Determinants | 行列式的性质

The following properties are essential for simplifying determinant calculations.

以下性质对于简化行列式计算至关重要。

If a matrix has a zero row or zero column, its determinant is zero.

若矩阵有一整行或一整列为零，则行列式为零。
Swapping two rows (or two columns) changes the sign of the determinant.

交换两行（或两列）会改变行列式的符号。
Adding a multiple of one row to another row does not change the determinant.

将某一行的倍数加到另一行上，行列式的值不变。
Multiplying a single row by a scalar k multiplies the determinant by k.

将某一行乘以标量 k，行列式也乘以 k。
det(AB) = det(A) × det(B) for two square matrices of the same order.

对于同阶方阵 A 和 B，有 det(AB) = det(A) × det(B)。
det(Aᵀ) = det(A), where Aᵀ is the transpose of A.

det(Aᵀ) = det(A)，其中 Aᵀ 是 A 的转置。

6. Geometric Interpretation (2D) | 几何意义（二维）

For a 2 × 2 matrix, the absolute value of the determinant represents the area of the parallelogram formed by the two column vectors of the matrix.

对于 2 × 2 矩阵，行列式的绝对值表示由矩阵的两个列向量所张成的平行四边形的面积。

If the column vectors are u = (a, c) and v = (b, d), then the area of the parallelogram with adjacent sides u and v is |ad − bc|.

若列向量为 u = (a, c) 和 v = (b, d)，则以 u 和 v 为邻边的平行四边形面积为 |ad − bc|。

Area = |det(A)|

The sign of the determinant indicates the orientation of the transformation: a positive determinant preserves orientation, while a negative determinant reverses it.

行列式的符号表示变换的定向：行列式为正表示保持定向，为负则表示反转定向。

7. Geometric Interpretation (3D) | 几何意义（三维）

For a 3 × 3 matrix, the absolute value of the determinant equals the volume of the parallelepiped formed by its three column vectors.

对于 3 × 3 矩阵，行列式的绝对值等于由它的三个列向量所张成的平行六面体的体积。

Volume = |det(A)|

If the three vectors lie in the same plane, the volume is zero, so the determinant is zero. This corresponds to linear dependence among the vectors.

若三个向量共面，则体积为零，因此行列式为零。这对应着向量之间的线性相关。

This idea connects directly to the scalar triple product in vector calculus: u · (v × w) = det[ u v w ].

这一思想直接联系到向量微积分中的标量三重积：u · (v × w) = det[ u v w ]。

8. Determinants and Matrix Inverses | 行列式与矩阵求逆

For a 2 × 2 matrix A = [ a b ; c d ], the inverse is given by:

对于二阶矩阵 A = [ a b ; c d ]，其逆矩阵为：

A⁻¹ = (1/(ad − bc)) × [ d −b ; −c a ]

Here ad − bc is the determinant. If the determinant is zero, the inverse does not exist.

其中 ad − bc 就是行列式。若行列式为零，则逆矩阵不存在。

For a 3 × 3 matrix, the inverse is A⁻¹ = (1/det(A)) × adj(A), where adj(A) is the adjugate matrix, the transpose of the cofactor matrix. In A-Level examinations, you may need to compute the inverse of a 2 × 2 matrix, and occasionally a 3 × 3 inverse using a calculator or by hand.

对于三阶矩阵，逆矩阵为 A⁻¹ = (1/det(A)) × adj(A)，其中 adj(A) 是伴随矩阵，即余子式矩阵的转置。在 A-Level 考试中，你可能需要计算二阶矩阵的逆，偶尔也需要手算或使用计算器求三阶逆矩阵。

9. Solving Linear Systems: Cramer’s Rule | 解线性方程组：克莱姆法则

Cramer’s rule uses determinants to solve systems of linear equations. For a system of two equations:

克莱姆法则利用行列式求解线性方程组。对于二元方程组：

a₁x + b₁y = c₁
a₂x + b₂y = c₂

Let D = a₁b₂ − a₂b₁ ≠ 0. Then:

设 D = a₁b₂ − a₂b₁ ≠ 0。则：

x = (c₁b₂ − c₂b₁) / D, y = (a₁c₂ − a₂c₁) / D

If D = 0, the system either has no solution or infinitely many solutions.

若 D = 0，则方程组可能无解或有无穷多解。

For three equations in three unknowns, Cramer’s rule extends naturally by replacing the first, second, or third column of the coefficient matrix with the constant vector and computing the corresponding determinants.

对于三元方程组，克莱姆法则可以自然推广：将系数矩阵的第一列、第二列或第三列替换为常数向量，然后计算对应的行列式。

10. Singular Matrices and Linear Dependence | 奇异矩阵与线性相关

A square matrix is singular if and only if its determinant is zero. This occurs when at least one row (or column) is a linear combination of the others.

方阵是奇异的当且仅当其行列式为零。这发生在至少一行（或一列）是其他行（或列）的线性组合时。

In the context of linear equations Ax = b, if det(A) = 0, the matrix A cannot be inverted, so either there is no unique solution.

在线性方程组 Ax = b 中，若 det(A) = 0，则矩阵 A 不可逆，因此不存在唯一解。

If a set of vectors has determinant zero, those vectors are linearly dependent; conversely, a non-zero determinant indicates linear independence.

若一组向量构成的行列式为零，则这些向量线性相关；反之，非零行列式表示线性无关。

11. Area and Volume Calculations | 面积与体积的计算

Determinants provide a direct method for finding areas and volumes without needing coordinate geometry formulas.

行列式提供了一种无需使用解析几何公式即可直接求面积和体积的方法。

For a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃), the area is:

对于顶点为 (x₁, y₁), (x₂, y₂), (x₃, y₃) 的三角形，其面积为：

Area = ½ × | x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂) |

This expression is exactly half the absolute value of the determinant of the 3 × 3 matrix formed by appending a column of ones:

该表达式恰好是由三行坐标加上一列 1 构成的三阶矩阵行列式绝对值的一半：

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

Similarly, the volume of a tetrahedron can be obtained from a 4 × 4 determinant, though this is beyond the core A-Level syllabus.

类似地，四面体的体积可以由四阶行列式求得，不过这超出了 A-Level 核心大纲的范围。

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

When computing determinants, always keep track of signs, especially in cofactor expansion. A common mistake is forgetting the alternating signs (−1)^(i+j).

计算行列式时，务必注意符号，尤其是在余子式展开中。常见错误是忘记交替符号 (−1)^(i+j)。

Check whether a matrix is singular before attempting to find its inverse.

在尝试求逆矩阵之前，先判断矩阵是否为奇异矩阵。
For 2 × 2 determinants, remember the order: ad − bc, not ac − bd.

对于二阶行列式，记住顺序：ad − bc，而不是 ac − bd。
Use row operations to simplify a 3 × 3 determinant before expanding, but remember that swapping rows changes the sign, and scaling a row multiplies the determinant.

在展开三阶行列式之前，可以使用行变换简化，但要记住交换行会改变符号，缩放行会乘以相应的倍数。
When a matrix has a factor common to all entries of a row, you can factor it out of the determinant.

当矩阵某一行所有元素含有公因子时，可以将该因子提到行列式外面。
Always verify your result using a calculator if permitted, or by substituting the solution back into the original equations.

如果允许，始终用计算器验证结果，或者将解代回原方程组进行检验。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Determinants | 行列式