Determinants | 行列式

📚 Determinants | 行列式

ns has a unique solution. This article covers the 2×2 and 3×3 cases, their geometric meaning, and typical exam applications.

在AQA A-Level数学（尤其是进阶数学模块）中，行列式给出一个数值，用来概括方阵的关键性质。行列式可以告诉我们矩阵是否可逆、它如何缩放面积或体积，以及线性方程组是否有唯一解。本文涵盖2×2和3×3行列式、其几何意义以及常见考试应用。

1. What is a determinant? | 什么是行列式？

rom the entries of a square matrix. It is written with vertical bars, for example det A or |A| x is singular and has no inverse. If det A ≠ 0, the matrix is non-singular and invertible.

行列式是由方阵的元素计算出的一个标量。它用竖线表示，例如 det A 或 |A|。只有方阵才有行列式。如果 det A = 0，该矩阵是奇异的，没有逆矩阵；如果 det A ≠ 0，该矩阵非奇异且可逆。

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2. 2×2 determinants | 2×2 行列式

minant is found by subtracting the product of the off-diagonal entries from the product of the main diagonal entries:

对于2×2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：

det A = ad − bc

det A = (3)(4) − (2)(1) = 12 − 2 = 10. Since the determinant is non-zero, A is invertible.

示例：若 A = [3 2; 1 4]，则 det A = (3)(4) − (2)(1) = 12 − 2 = 10。因为行列式非零，所以 A 可逆。

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3. 3×3 determinants | 3×3 行列式

n method is expansion by the first row. For A = [a b c; d e f; g h i], the determinant is:

3×3行列式可以沿任意行或列使用余子式展开。常用方法是按第一行展开。对于 A = [a b c; d e f; g h i]，行列式为：

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

ained by deleting the row and column of the entry being used. The signs alternate +, −, + across the first row.

每个2×2行列式是删除当前元素所在行和列后得到的余子式。符号沿第一行按 +、−、+ 交替。

+

−

+

7) = 1(45 − 48) − 2(36 − 42) + 3(32 − 35) = −3 + 12 − 9 = 0. Hence the matrix is singular.

示例： ×8) − 2(4×9 − 6×7) + 3(4×8 − 5×7) = 1(45 − 48) − 2(36 − 42) + 3(32 − 35) = −3 + 12 − 9 = 0。因此该矩阵是奇异的。

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4. Properties of determinants | 行列式的性质

det(I) = 1 for any identity matrix.

任何单位矩阵的行列式 det(I) = 1。

AB) = det(A) × det(B), so the determinant of a product is the product of the determinants.

det(AB) = det(A) × det(B)，即矩阵乘积的行列式等于各行列式的乘积。

det(Aᵀ) = det(A), where Aᵀ is the transpose of A.

det(Aᵀ) = det(A)，其中 Aᵀ 是 A 的转置。

Interchanging 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。

Multiplying one row by k multiplies the determinant by k.

将一行乘以 k，行列式也乘以 k。

If one row is a multiple of another row, then det A = 0.

如果某一行是另一行的倍数，则 det A = 0。

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5. Geometric meaning in 2D | 二维几何意义

of det A gives the area scale factor of the linear transformation represented by A. If det A is negative, the transformation reverses orientation.

对于2×2矩阵 A，|det A| 给出该矩阵所表示线性变换的面积缩放因子。如果 det A 为负，变换会反转方向。

Area of image = |det A| × original area

mple: The matrix [2 0; 0 3] has det = 6. It maps the unit square to a rectangle of area 6.

示例：矩阵 [2 0; 0 3] 的行列式为6。它将单位正方形映射为面积为6的矩形。

n 2D, the transformation collapses the plane onto a line or a point, so area becomes zero.

如果在二维中 det A = 0，该变换将平面压缩到一条直线或一个点，因此面积变为零。

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6. Geometric meaning in 3D | 三维几何意义

In 3D, |det A| matrix also represents the scalar triple product of its column vectors, which equals the volume of the parallelepiped spanned by them.

在三维中，|det A| 是变换的体积缩放因子。3×3矩阵的行列式也表示其列向量的标量三重积，等于它们张成的平行六面体体积。

Volume of image = |det A| × original volume

If the column vectors are coplanar, the parallelepiped has zero volume, so det A = 0 and the matrix is singular.

如果列向量共面，平行六面体体积为零，因此 det A = 0，矩阵奇异。

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7. Determinants and inverses | 行列式与逆矩阵

f and only if det A ≠ 0. The inverse of a 2×2 matrix can be written using the determinant:

方阵 A 有逆矩阵当且仅当 det A ≠ 0。2×2矩阵的逆可以用行列式表示：

A⁻¹ = (1 / det A) × [ d −b; −c a ]

xample, if A = [3 2; 1 4], det A = 10, so A⁻¹ = (1/10)[4 −2; −1 3] = [0.4 −0.2; −0.1 0.3].

例如，若 A = [3 2; 1 4]，det A = 10，因此 A⁻¹ = (1/10)[4 −2; −1

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