📚 Determinants | 行列式
In linear algebra, the determinant is a scalar value that can be computed from the elements of a square matrix and encodes essential information such as invertibility, volume scaling, and linear dependence. Understanding determinants is a central part of IB Mathematics, connecting algebraic manipulation with geometric interpretation. This article provides a thorough exploration of determinants, from basic computation to advanced properties and real-world applications.
在线性代数中，行列式是一个可以从方阵元素计算出的标量值，蕴含了矩阵可逆性、体积缩放和线性相关性等关键信息。理解行列式是 IB 数学的核心部分，它将代数操作与几何解释紧密联系起来。本文从基本计算到高级性质及实际应用，对行列式进行全面探讨。
1. Definition of a Determinant | 行列式的定义
umber to every square matrix. For an n × n matrix A, the determinant is denoted by det(A), |A| ant provides a direct test for invertibility: if det(A) = 0, the matrix A is singular (non‑invertible); if det(A) ≠ 0, A is non‑ pt originates from solving systems of linear equations and has powerful geometric meaning.
行列式是一个给每个方阵分配唯一实数值的函数。对于 n × n 矩阵 A，行列式记为 det(A)、|A| 或用垂直线包围矩阵元素来表示。行列式的值提供了可逆性的直接检验：若 det(A) = 0，则矩阵 A 是奇异的（不可逆）；若 det(A) ≠ 0，则 A 是非奇异的（可逆）。这一概念源于求解线性方程组，并具有强大的几何意义。
The definition is built recursively, starting from the simplest non‑ r matrices, the determinant can be defined via cofactor expansion or by its properties. No matter the size, the determinant remains a single number that captures the “scaling factor” of the linear transformation represented by the matrix.
定义是递归构建的，从最简单的非平凡情况——2 × 2 矩阵开始。对于更大的矩阵，行列式可以通过余子式展开或其性质来定义。无论矩阵多大，行列式始终是反映该矩阵所表示线性变换“缩放因子”的一个单一数字。
2. Determinant of a 2×2 Matrix | 2×2 矩阵的行列式
determinant is calculated as det(A) = a₁₁a₂₂ − a₁₂a₂₁. This formula is often remembered as “product of the main diagonal minus product of the off‑diagonal.”
给定 2 × 2 矩阵 A = [a₁₁ a₁₂; a₂₁ a₂₂]，行列式计算为 det(A) = a₁₁a₂₂ − a₁₂a₂₁。这个公式常记为“主对角线乘积减去副对角线乘积”。
det(A) = | a₁₁ a₁₂ | = a₁₁a₂₂ − a₁₂a₂₁
| a₂₁ a₂₂ |
For example, consider A = [2 3; 1 4]. Then det(A) = (2)(4) − (3)(1) = 8 − 3 = 5. A non‑ such as [2 3; 4 6], the determinant would be (2)(6) − (3)(4) = 0, indicating singularity.
例如，考虑 A = [2 3; 1 4]。则 det(A) = (2)(4) − (3)(1) = 8 − 3 = 5。非零行列式表明该矩阵可逆，且行（与列）线性无关。如果矩阵的两行成比例，例如 [2 3; 4 6]，那么行列式 (2)(6) − (3)(4) = 0，表示矩阵奇异。
3. Determinant of a 3×3 Matrix (Sarrus' Rule) | 3×3 矩阵的行列式（萨鲁斯法则）
For a 3 × 3 matrix A = [a₁₁ a₁₂ a₁₃; a₂₁ a₂₂ a₂₃; a₃₁ a₃₂ a₃₃], a quick computation
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