Determinants of 2×2 and 3×3 Matrices | 二阶与三阶行列式

📚 Determinants of 2×2 and 3×3 Matrices | 二阶与三阶行列式

Determinants are fundamental to matrix algebra. For a square matrix, the determinant is a single number that reveals whether the matrix is invertible, how linear transformations scale areas or volumes, and even provides a direct method for solving systems of linear equations. This article covers everything you need for AQA A-Level Mathematics: the determinant of a 2×2 matrix, the determinant of a 3×3 matrix, their properties, and their applications.

行列式是矩阵代数中的核心概念。对于任意方阵，行列式是一个标量，它能够揭示矩阵是否可逆，反映线性变换对面积或体积的缩放程度，并且提供了求解线性方程组的一种直接方法。本文针对 AQA A-Level 数学大纲，系统讲解二阶行列式、三阶行列式、行列式的性质及其应用。

1. What is a Determinant? | 什么是行列式？

A determinant is a scalar value associated with a square matrix. It is written as det(A) or |A|. For a 1×1 matrix A = [a], the determinant is simply a. For larger matrices, the determinant is computed by a recursive process that ultimately reduces to arithmetic on the entries of the matrix.

行列式是与方阵相关联的一个标量值，通常记作 det(A) 或 |A|。对于 1×1 矩阵 A = [a]，其行列式就是 a 本身。对于更高阶的矩阵，行列式通过递归的展开过程计算，最终化为矩阵元素的算术运算。

The determinant has several important interpretations. In algebra, it determines whether a matrix is invertible. In geometry, the absolute value of the determinant gives the scale factor by which an area or volume is changed under a linear transformation. In equation solving, determinants appear directly in Cramer’s rule.

行列式具有多种重要的含义。在代数中，它决定矩阵是否可逆；在几何中，其绝对值表示线性变换对面积或体积的放大倍数；在解方程中，行列式直接出现在克莱默法则中。

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

For a 2×2 matrix, the determinant is the difference of the products of its two diagonals. If we write the matrix as

对于二阶矩阵，行列式等于主对角线乘积减去副对角线乘积。若矩阵写成

A = [a b; c d]

then the determinant is ad − bc. This is the most basic formula and appears in many exam questions.

则其行列式为 ad − bc。这是最基本的公式，也是许多考试问题的起点。

Example: Let A = [2 3; 4 5]. Then det(A) = 2×5 − 3×4 = 10 − 12 = −2.

例如：设 A = [2 3; 4 5]，则 det(A) = 2×5 − 3×4 = 10 − 12 = −2。

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

For a 3×3 matrix, the determinant can be found by expanding along any row or column. This method uses minors and cofactors. Consider the general 3×3 matrix

对于三阶矩阵，行列式可以通过按任意一行或一列展开来计算。该方法用到余子式和代数余子式。考虑一般三阶矩阵

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

Expanding along the first row gives

按第一行展开得到

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

Notice that the signs alternate: first a positive, then b negative, then c positive. The pattern of signs for the first row is +, −, + because of the cofactor sign (−1)^i+j.

注意符号交替：第一项 a 为正，第二项 b 为负，第三项 c 为正。第一行符号模式为 +、−、+，这是由代数余子式的符号 (−1)^i+j 决定的。

4. The Rule of Sarrus | 萨鲁斯法则

The Rule of Sarrus is a shortcut method for finding the determinant of a 3×3 matrix only. It is not valid for 4×4 or larger matrices. To use it, copy the first two columns to the right of the matrix, then multiply along the diagonals.

萨鲁斯法则是仅适用于三阶行列式的快捷方法，对于四阶及以上矩阵不成立。使用方法是把矩阵的前两列写在右边，然后沿对角线相乘。

Suppose the matrix is written as:

假设矩阵写为：

a b c a b
d e f d e
g h i g h

Then the determinant is the sum of the three downward diagonals products minus the sum of the three upward diagonal products:

行列式等于三条下降对角线乘积之和减去三条上升对角线乘积之和：

|A| = (aei + bfg + cdh) − (ceg + afh + bdi)

This formula gives the same result as the cofactor expansion. Use it when you are confident with the method; the cofactor expansion is always dependable.

使用该公式会得到与按行展开相同的结果。如果你对步骤有把握，可以使用萨鲁斯法则；而按行展开永远是可靠的方法。

5. Properties of Determinants | 行列式的性质

Several properties of determinants are essential for simplifying calculations and for proofs. Each property also applies to any n×n square matrix.

行列式的若干重要性质能帮助简化计算，也常用于证明题之中。下面这些性质适用于任意 n×n 方阵。

det(I) = 1 — The determinant of the identity matrix is 1. | 单位矩阵的行列式为 1。
det(Aᵀ) = det(A) — The determinant of a transpose equals the determinant of the original matrix. | 转置矩阵与原矩阵的行列式相等。
det(AB) = det(A)det(B) — For two square matrices of the same size. | 对于两个同阶方阵，AB 的行列式等于 det(A) 与 det(B) 的乘积。
det(kA) = kⁿdet(A) — For an n×n matrix, multiplying every entry by k multiplies the determinant by kⁿ. | 对于一个 n×n 矩阵，若所有元素乘以 k，则行列式乘以 kⁿ。
If two rows (or columns) are equal or proportional, det(A) = 0. | 若矩阵有两行（或两列）相等或成比例，则行列式为零。
Swapping two rows (or columns) multiplies the determinant by −1. | 交换两行（或两列），行列式变号。

These properties can often reduce a complicated determinant to a simpler form. For instance, if one row is a multiple of another, the determinant is instantly zero.

这些性质往往能把复杂的行列式化简。例如，若某一行是另一行的倍数，则可以立刻判断行列式为零。

6. Singular and Non-Singular Matrices | 奇异矩阵与非奇异矩阵

A square matrix is called singular if its determinant is zero, and non-singular (or invertible) if its determinant is non-zero. This distinction is crucial in linear algebra.

行列式为零的方阵称为奇异矩阵；行列式不为零的方阵称为非奇异矩阵或可逆矩阵。这一区分在线性代数中极为关键。

For a non-singular matrix, an inverse exists. For a singular matrix, no inverse exists, and the associated system of linear equations either has no solution or infinitely many solutions.

非奇异矩阵存在逆矩阵；奇异矩阵不存在逆矩阵，且其对应的线性方程组可能无解或有无穷多解。

Example: The matrix [1 2; 2 4] has determinant 1×4 − 2×2 = 0, so it is singular. The rows are proportional, and no unique solutions can be guaranteed.

例如：矩阵 [1 2; 2 4] 的行列式为 1×4 − 2×2 = 0，所以它是奇异矩阵。其两行成比例，因此无法保证唯一解。

7. The Inverse of a Matrix using Determinants | 利用行列式求逆矩阵

The inverse of a 2×2 matrix is directly linked to its determinant. For A = [a b; c d], if det(A) ≠ 0, then

二阶矩阵的逆矩阵直接与行列式相关。对于 A = [a b; c d]，若 det(A) ≠ 0，则

A⁻¹ = 1/det(A) [d −b; −c a]

In words: swap the two diagonal entries, change the signs of the off-diagonal entries, and divide every entry by the determinant.

口诀是：主对角线交换位置，副对角线改变符号，然后每个元素除以行列式。

Example: Let A = [2 1; 5 3]. Then det(A) = 2×3 − 1×5 = 1, so A⁻¹ = [3 −1; −5 2].

例如：设 A = [2 1; 5 3]，则 det(A) = 2×3 − 1×5 = 1，所以 A⁻¹ = [3 −1; −5 2]。

For a 3×3 matrix, the inverse is given by the adjugate formula: A⁻¹ = (1/det(A)) adj(A), where adj(A) is the transpose of the matrix of cofactors. This is more advanced but follows the same principle.

对于三阶矩阵，逆矩阵由伴随矩阵公式给出：A⁻¹ = (1/det(A)) adj(A)，其中 adj(A) 是代数余子式矩阵的转置。这个公式更高级一些，但原理相同。

8. Solving Linear Equations with Determinants | 利用行列式求解线性方程组

Determinants can be used to solve a system of linear equations via Cramer’s rule. For a 2×2 system,

行列式可以通过克莱默法则求解线性方程组。对于二元方程组

a₁x + b₁y = c₁
a₂x + b₂y = c₂

Let D be the determinant of the coefficient matrix:

设 D 是系数矩阵的行列式：

D = |a₁ b₁; a₂ b₂| = a₁b₂ − a₂b₁

Then replace the x-column with the constants c₁ and c₂:

然后用常数项替换 x 列：

Dₓ = |c₁ b₁; c₂ b₂|

and similarly for the y-column:

同理替换 y 列：

Dᵧ = |a₁ c₁; a₂ c₂|

If D ≠ 0, the unique solution is x = Dₓ/D and y = Dᵧ/D. This extends to 3×3 systems by replacing one column at a time.

若 D ≠ 0，则有唯一解 x = Dₓ/D，y = Dᵧ/D。该方法可推广到三元方程组，只需每次替换一列。

9. Geometric Interpretation: Area and Volume | 几何意义：面积与体积

The determinant has a natural geometric meaning. For a 2×2 matrix, the absolute value of the determinant equals the area of the parallelogram formed by its column vectors. If the determinant is negative, the transformed orientation is reversed.

行列式有自然的几何意义。对于二阶矩阵，行列式的绝对值等于由两个列向量构成的平行四边形的面积。若行列式为负，则表示变换后的方向发生反转。

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

对于三阶矩阵，行列式的绝对值等于由三个列向量构成的平行六面体的体积。同样，符号表示三个向量的方向性。

Under a linear transformation with matrix A, areas are multiplied by |det(A)|. For 3D transformations, volumes are multiplied by |det(A)|. This is why a determinant of zero collapses the shape to a line or plane.

在线性变换 A 作用下，面积被乘以 |det(A)|。对于三维变换，体积被乘以 |det(A)|。这也是为什么行列式为零时，图形会被压缩成一条直线或一个平面。

10. Summary of Key Formulas | 关键公式总结

Matrix Size \| 矩阵阶数	Formula \| 公式
1×1 [a]	det = a
2×2 [a b; c d]	det = ad − bc
3×3	det = a(ei − fh) − b(di − fg) + c(dh − eg) (Sarrus: (aei + bfg + cdh) − (ceg + afh + bdi))

Remember: if det(A) = 0, A is singular; if det(A) ≠ 0, A is invertible. Use Cramer’s rule or the inverse matrix method to solve simultaneous equations.

请记住：若 det(A) = 0，则 A 是奇异矩阵；若 det(A) ≠ 0，则 A 可逆。可以用克莱默法则或逆矩阵法求解线性方程组。

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Determinants of 2×2 and 3×3 Matrices | 二阶与三阶行列式