📚 Cramer's Rule | 克莱姆法则
Cramer's rule provides an elegant determinant-based formula for solving a system of n linear equations in n unknowns, provided the coefficient matrix is invertible. It expresses each variable as a ratio of two determinants: the determinant of the original coefficient matrix and the determinant of a modified matrix where one column is replaced by the constant vector. Named after the Swiss mathematician Gabriel Cramer, this rule is particularly useful for small systems and for theoretical analysis in linear algebra.
克莱姆法则提供了一种基于行列式的优美公式，用于求解 n 个方程 n 个未知数的线性方程组，要求系数矩阵可逆。它将每个未知数表示为两个行列式之比：原系数矩阵的行列式，以及将其中一列替换为常数向量后所得矩阵的行列式。该法则以瑞士数学家加布里埃尔·克莱姆命名，在小型方程组求解和线性代数理论分析中尤为实用。
1. What is Cramer's Rule? | 什么是克莱姆法则？
atrix, X is the column vector of unknowns, and B is the column vector of constants, Cramer's rule states that the i-th unknown xᵢ (i = 1, 2, … 0. The matrix Aᵢ is obtained by replacing the i-th column of A with the constant vector B.
对于写成矩阵形式 AX = B 的线性方程组，其中 A 是 n × n 系数矩阵，X 是未知数列向量，B 是常数列向量，克莱姆法则指出，第 i 个未知数 xᵢ（i = 1, 2, …, n）由 xᵢ = det(Aᵢ) / det(A) 给出，前提是 det(A) ≠ 0。矩阵 Aᵢ 是把 A 的第 i 列替换为常数向量 B 得到的矩阵。
2. When Can You Use It? | 适用条件
Cramer' stem either has no unique solution (inconsistent or infinitely many solutions), and Cramer' ractical for 2 × 2 and 3 × 3 systems due to the computational cost of larger determinants.
克莱姆法则要求系数矩阵为方阵（方程个数等于未知数个数）且行列式非零。若 det(A) = 0，方程组要么无解，要么有无穷多解（不一致或无唯一解），此时克莱姆法则不适用。该法则还假设所有方程均为线性方程，常数为实数（或复数）。由于高阶行列式计算量大，该法则最适用于 2 × 2 和 3 × 3 的方程组。
3. 2×2 System Step-by-Step | 2×2 方程组步骤详解
Consider the system:
考虑以下方程组：
a₁x + b₁y = c₁
a₂x + b₂y = c₂
Write the coefficient matrix A and constant vector B. Then compute Δ = det(A) = a₁b₂ – a₂b₁. Next, form Δ₁ by replacing the first column (coefficients of x) with B: Δ₁ = c₁b₂ – c₂b₁. Form Δ₂ by replacing the second column (coefficients of y) with B: Δ₂ = a₁c₂ – a₂c₁. The unique solution is x = Δ₁ / Δ, y = Δ₂ / Δ.
写出系数矩阵 A 和常数向量 B。然后计算 Δ = det(A) = a₁b₂ – a₂b₁。接着，将第一列（x 的系数）替换为 B 得到 Δ₁：Δ₁ = c₁b₂ – c₂b₁。将第二列（y 的系数）替换为 B 得到 Δ₂：Δ₂ = a₁c₂ – a₂c₁。唯一解为 x = Δ₁ / Δ，y = Δ₂ / Δ。
4. Worked Example: 2×2 | 2×2 实例演示
Solve the system using Cramer's rule:
用克莱姆法则求解方程组：
2x + 3y = 5
4x + y = 11
Here a₁=2, b₁=3, c₁=5; a₂=4, b₂=1, c₂=11. Compute Δ = 2×1 – 4×3 = 2 – 12 = –10. Then Δ₁ = 5×1 – 11×3 = 5 – 33 = –28. Δ₂ = 2×11 – 4×5 = 22 – 20 = 2. Thus x = (–28)/(–10) = 2.8, y = 2/(–10) = –0.2. Check: 2(2.8) + 3(–0.2) = 5.6 – 0.6 = 5; 4(2.8) + (–0.2) = 11.2 – 0.2 = 11.
这里 a₁=2, b₁=3, c₁=5; a₂=4, b₂=1, c₂=11。计算 Δ = 2×1 – 4×3 = 2 – 12 = –10。然后 Δ₁ = 5×1 – 11×3 = 5 – 33 = –28。Δ₂ = 2×11 – 4×5 = 22 – 20 = 2。因此 x = (–28)/(–10) = 2.8，y = 2/(–10) = –0.2。验证：2(2.8) + 3(–0.2) = 5.6 – 0.6 = 5；4(2.8) + (–0.2) = 11.2 – 0.2 = 11。
5. 3×3 System & Determinant Expansion | 3×3 方程组与行列式展开
Δ₁/Δ, y = Δ₂/Δ, z = Δ₃/Δ. A 3×3 determinant can be expanded using the first row or Sarrus' rule: det(A) = a₁(b₂c₃ – b₃c₂) – b₁(a₂c₃ – a₃c₂) + c₁(a₂b₃ – a₃b₂) when A has rows (a₁,b₁,c₁), etc.
对于 3×3 方程组 AX = B，令 Δ = det(A)。若 Δ ≠ 0，则将 A 的第一、第二、第三列分别替换为 B，算出 Δ₁、Δ₂、Δ₃。然后 x = Δ₁/Δ，y = Δ₂/Δ，z = Δ₃/Δ。3×3 行列式可用第一行展开或萨鲁斯法则计算：当 A 的行为 (a₁,b₁,c₁) 等，det(A) = a₁(b₂c₃ – b₃c₂) – b₁(a₂c₃ – a₃c₂) + c₁(a₂b₃ – a₃b₂)。
6. Worked Example: 3×3 | 3×3 实例演示
Solve the system:
求解方程组：
x + 2y + z = 4
2x – y + 3z = 5
–x + y – 2z = –3
Matrix A has rows (1,2,1), (2,–1,3), (–1,1,–2). Δ = 1( (–1)(–2) – (3)(1) ) – 2( (2)(–2) – (3)(–1) ) + 1( (2)(1) – (–1)(–1) ) = 1(2 – 3) – 2(–4 + 3) + 1(2 – 1) = –1 – 2(–1) + 1 = –1 + 2 + 1 = 2. Δ₁ replaces first column: rows (4,2,1), (5,–1,3), (–3,1,–2). Δ₁ = 4( (–1)(–2) – 3·1 ) – 2( 5(–2) – 3·(–3) ) + 1( 5·1 – (–1)(–3) ) = 4(2–3) –2(–10 +9) +1(5 –3) = –4 –2(–1) +2 = –4+2+2 = 0. Δ₂ replaces second column: rows (1,4,1), (2,5,3), (–1,–3,–2). Δ₂ = 1(5(–2) – 3(–3)) – 4(2(–2) – 3(–1)) + 1(2(–3) – 5(–1)) = 1(–10+9) –4(–4+3) +1(–6+5) = –1 –4(–1) –1 = –1+4–1 = 2. Δ₃ replaces third column: rows (1,2,4), (2,–1,5), (–1,1,–3). Δ₃ = 1(–1(–3) –5·1) –2(2(–3) –5(–1)) +4(2·1 –(–1)(–1)) = 1(3–5) –2(–6+5) +4(2–1) = –2 –2(–1) +4 = –2+2+4 = 4. Thus x = 0/2 = 0, y = 2/2 = 1, z = 4/2 = 2.
矩阵 A 的行为 (1,2,1), (2,–1,3), (–1,1,–2)。Δ = 1( (–1)(–2) – (3)(1) ) – 2( (2)(–2) – (3)(–1) ) + 1( (2)(1) – (–1)(–1) ) = 1(2–3) –2(–4+3) +1(2–1) = –1 –2(–1)+1 = –1+2+1 = 2。Δ₁ 替换第一列：行为 (4,2,1), (5,–1,3), (–3,1,–2)。Δ₁ = 4( (–1)(–2) – 3·1 ) –2( 5(–2) – 3·(–3) ) +1( 5·1 – (–1)(–3) ) = 4(2–3) –2(–10+9) +1(5–3) = –4 –2(–1)+2 = –4+2+2 = 0。Δ₂ 替换第二列：行为 (1,4,1), (2,5,3), (–1,–3,–2)。Δ₂ = 1(5(–2) – 3(–3)) –4(2(–2) – 3(–1)) +1(2(–3) – 5(–1)) = 1(–10+9) –4(–4+3) +1(–6+5) = –1 –4(–1) –1 = –1+4–1 = 2。Δ₃ 替换第三列：行为 (1,2,4), (2,–1,5), (–1,1,–3)。Δ₃ = 1(–1(–3) –5·1) –2(2(–3) –5(–1)) +4(2·1 –(–1)(–1)) = 1(3–5) –2(–6+5) +4(2–1) = –2 –2(–1)+4 = –2+2+4 = 4。因此 x = 0/2 = 0，y = 2/2 = 1，z = 4/2 = 2。
7. Geometric Interpretation | 几何意义
In two dimensions, the absolute value of det(A) represents the area of the parallelogram spanned by the column vectors of A. The ratio Δ₁/Δ can be seen as comparing a ‘distorted’ area to the original area, effectively giving the coordinate that scales the basis vector. In three dimensions, det(A) is the volume of the parallelepiped formed by the column vectors, and Cramer's rule expresses each coordinate as a volume ratio. The non-zero determinant condition guarantees that the vectors are linearly independent and span the space.
在二维中，det(A) 的绝对值表示 A 的列向量所张成的平行四边形的面积。比值 Δ₁/Δ 可看作将“扭曲”后的面积与原面积进行比较，从而给出缩放基向量的坐标。在三维中，det(A) 是由列向量构成的平行六面体的体积，克莱姆法则将每个坐标表示为体积之比。行列式非零的条件保证了向量线性无关且张成整个空间。
8. Connection to Matrix Inverse | 与逆矩阵的联系
Cramer' f adj(A))·B / det(A), precisely the determinant ratio in Cramer’s rule. This shows why the 'replaced column' nts appear: they are the dot products of B with the cofactors of the corresponding column.
克莱姆法则与矩阵的逆密切相关。对于可逆矩阵 A，A⁻¹ = (1/det(A)) adj(A)，其中 adj(A) 是伴随矩阵，其元素为余子式。解 X = A⁻¹B 将每个 xᵢ 表示为 (adj(A) 的行)·B / det(A)，这正是克莱姆法则中的行列式之比。这解释了为何会出现“替换列”行列式：它们是 B 与对应列的余子式的点积。
9. Cramer's Rule in IB Mathematics | 在 IB 数学中的考查
In the IB Diploma Programme, Cramer' ret the meaning of a zero determinant, and understand the connection with matrix invertibility. Questions often combine Cramer's rule with geometric interpretations or proofs related to unique solutions.
在 IB 文凭课程中，克莱姆法则出现在 AA（分析与方法）高水平课程大纲中，属于线性方程组与行列式主题。学生应能使用该法则求 2×2 和 3×3 方程组的唯一解，解释行列式为零的意义，并理解与矩阵可逆性的联系。考题常将克莱姆法则与几何意义或关于唯一解的证明结合在一起。
10. Common Mistakes & Tips | 常见错误与提示
3 determinants; and using the rule for non-square systems. Tips: always label columns with the corresponding variables; compute Δ first— check for consistency or parameters; re-check your determinant expansions; and use Cramer' rule as a check against other methods like elimination or matrix inverse on a calculator.
常见错误包括：计算出行列式 Δ 但忘记检查其是否为零；混淆了每个变量应替换哪一列；3×3 行列式展开时出现符号错误；以及在非方阵方程组中错误使用该法则。提示：始终用对应的变量标记各
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