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Determinants | 行列式

2026-08-21 20:29 · 本文部分内容由AI辅助创作

📚 Determinants | 行列式

A determinant is a special scalar quantity that is calculated from the entries of a square matrix. In AQA A-Level Further Mathematics, determinants are essential for testing whether a matrix is invertible, solving simultaneous equations, and understanding how linear transformations change area or volume.

行列式是由方阵元素计算出的一个特殊标量。在 AQA A-Level 进阶数学中，行列式是判断矩阵是否可逆、求解线性方程组、以及理解线性变换如何改变面积或体积的重要工具。

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

If A is the 2 × 2 matrix with entries a, b, c, d, written as

如果 A 是元素为 a、b、c、d 的 2 × 2 矩阵，写作

A = [[a, b], [c, d]]

then its determinant is

那么它的行列式为

det(A) = |A| = ad − bc

This formula is obtained by multiplying the top-left entry by the bottom-right entry and subtracting the product of the top-right and bottom-left entries.

这个公式的计算方法是：左上元素乘右下元素，再减去右上元素与左下元素的乘积。

Example: Find det(A) for A = [[3, 5], [2, 7]].

示例：求 A = [[3, 5], [2, 7]] 的行列式。

det(A) = (3)(7) − (5)(2) = 21 − 10 = 11

Since the determinant is non-zero, the matrix is invertible.

由于行列式不为零，该矩阵是可逆的。

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

For a 3 × 3 matrix, the determinant can be found by expanding along the first row. If

对于 3 × 3 矩阵，可以沿第一行展开求行列式。如果

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

then the determinant is

则行列式为

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

st row and the determinant of the 2 × 2 minor matrix that remains after deleting the entry's row and column. The signs alternate +, −, +.

每一项都是第一行元素与去掉该元素所在行和列后剩下的 2 × 2 子式行列式的乘积。符号按 +、−、+ 交替。

Example: Find det(A) for A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]].

示例：求 A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]] 的行列式。

det(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 determinant is 0, so the matrix is singular.

该行列式为 0，因此该矩阵是奇异矩阵。

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3. Geometric Meaning: Area and Volume Scale Factor | 几何意义：面积与体积缩放因子

the plane, the absolute value of det(A) is the factor by which the area is scaled. If det(A) is negative, the transformation also reverses orientation.

如果 2 × 2 矩阵 A 对平面图形进行变换，|det(A)| 就是面积缩放因子。如果 det(A) 为负，该变换还会改变方向。

For a 3 × 3 matrix, |det(A)| is the volume scale factor under the linear transformation.

对于 3 × 3 矩阵，|det(A)| 是线性变换下的体积缩放因子。

the unit square to a rectangle. Since det(A) = 6, the image has area 6 times the original.

示例：变换矩阵 A = [[2, 0], [0, 3]] 将单位正方形映射为矩形。由于 det(A)=6，像的面积是原面积的 6 倍。

transformation collapses the shape onto a line or a point, so area or volume becomes zero.

如果 det(A)=0，变换会将图形压缩到一条直线或一个点上，因此面积或体积变为零。

4. Key Properties of Determinants | 行列式的主要性质

These properties are frequently tested and can simplify calculations.

以下性质经常考到，也能简化计算。