Cramer's Rule Calculator
用克拉瑪法則解二元或三元一次聯立方程式。輸入係數後可見行列式逐步計算、矩陣動畫、幾何意義與完整解。
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Cramer's Rule Calculator
克拉瑪法則計算機（Cramer's Rule Calculator）使用行列式求解 2 元或 3 元線性方程組。輸入係數矩陣與常數向量，即可獲得完整解答，包含逐步行列式計算、以動畫呈現欄位替換的矩陣視覺化，以及 2×2 方程組的幾何解釋圖。克拉瑪法則是線性代數中的基礎技巧，將每個變數表示為兩個行列式的比值。
What Is Cramer's Rule?
克拉瑪法則是線性代數中的一項定理，只要方程組的方程式數量與未知數數量相同且有唯一解，就能提供求解該線性方程組的明確公式。此法則以瑞士數學家加百列・克拉瑪（Gabriel Cramer，1704–1752）命名，利用行列式將每個變數表示為比值：
$$x_i = \frac{D_i}{D}$$
其中 \(D\) 是係數矩陣的行列式，\(D_i\) 是將係數矩陣的第 \(i\) 欄替換為常數向量後所形成的行列式。
Key Concepts
Determinant
A scalar value computed from a square matrix that indicates whether the system has a unique solution.
Column Replacement
Replace one column of the coefficient matrix with the constants vector to form each D_i.
Unique Solution
Exists when D ≠ 0. Each variable equals D_i / D.
Singular Case
When D = 0, the system has either no solution or infinitely many.
Cramer's Rule Formulas
For a 2×2 System
Given the system:
$$a_1x + b_1y = c_1$$ $$a_2x + b_2y = c_2$$
| Determinant | Formula | Description |
|---|---|---|
| \(D\) | \(\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix} = a_1 b_2 - b_1 a_2\) | Coefficient matrix determinant |
| \(D_x\) | \(\begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix} = c_1 b_2 - b_1 c_2\) | Replace x-column with constants |
| \(D_y\) | \(\begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix} = a_1 c_2 - c_1 a_2\) | Replace y-column with constants |
Solution: \(x = D_x / D\), \(y = D_y / D\)
For a 3×3 System
3×3 矩陣的行列式是透過沿第一列進行餘因子展開來計算的。每個 \(D_i\) 是透過將對應欄替換為常數向量而形成，解為 \(x_i = D_i / D\)。
When Does Cramer's Rule Work?
| Condition | D Value | Result |
|---|---|---|
| Unique solution | D ≠ 0 | Each variable = D_i / D |
| No solution (inconsistent) | D = 0, some D_i ≠ 0 | Lines/planes are parallel |
| Infinitely many solutions | D = 0, all D_i = 0 | Equations are dependent |
Cramer's Rule vs. Other Methods
| Method | Best For | Limitation |
|---|---|---|
| Cramer's Rule | Small systems (2×2, 3×3), exact symbolic solutions | Slow for large systems (n! complexity) |
| Gaussian Elimination | General systems, large matrices | No closed-form formula |
| Matrix Inverse | Multiple right-hand sides | Requires D ≠ 0, expensive to compute |
| LU Decomposition | Repeated solving, numerical stability | More complex to implement |
How to Use the Cramer's Rule Calculator
- 選擇方程組大小：依您的方程式與未知數數量，選擇 2×2 或 3×3。
- 輸入係數：填入左側的係數矩陣。每一列對應一個方程式，每一欄對應一個變數（x、y、z）。
- 輸入常數：填入右側的常數向量（即每個方程式的右邊）。
- 點擊求解：計算機會計算所有行列式（D、D_x、D_y，以及可能的 D_z），判斷解的類型，並以動畫矩陣視覺化呈現逐步計算過程。
Real-World Applications
| Field | Application | Example |
|---|---|---|
| Engineering | Circuit analysis (Kirchhoff's laws) | Finding currents in a resistor network |
| Economics | Market equilibrium | Supply and demand intersection |
| Physics | Force balance | Finding reaction forces in statics |
| Chemistry | Balancing equations | Stoichiometric coefficients |
| Computer Graphics | Coordinate transformations | Line/plane intersection points |
FAQ
What is Cramer's rule?
Cramer's rule is a method for solving a system of linear equations using determinants. For each variable, you replace its column in the coefficient matrix with the constants vector and divide the resulting determinant by the main determinant. It works when the coefficient matrix has a nonzero determinant.
When does Cramer's rule fail?
Cramer's rule fails when the determinant of the coefficient matrix is zero. This means the system either has no solution (inconsistent — the equations describe parallel lines or planes) or infinitely many solutions (dependent — the equations are redundant). In such cases, other methods like Gaussian elimination are needed.
What is the formula for Cramer's rule in a 2×2 system?
For the system a1*x + b1*y = c1, a2*x + b2*y = c2: x = Dx/D and y = Dy/D, where D = a1*b2 - b1*a2 is the determinant of the coefficient matrix, Dx replaces the x-column with the constants, and Dy replaces the y-column with the constants.
Can Cramer's rule solve systems larger than 3×3?
Cramer's rule can theoretically solve any n×n system, but it becomes computationally expensive for large systems because it requires computing n+1 determinants, each of size n×n. For systems larger than 3×3, methods like Gaussian elimination or LU decomposition are far more efficient in practice.
What does a zero determinant mean geometrically?
For a 2×2 system, a zero determinant means the two lines are parallel (no solution) or coincident (infinitely many solutions). For a 3×3 system, it means the three planes do not intersect at a single point — they may be parallel, intersect along a line, or all coincide in a plane.
引用此內容、頁面或工具為：
"Cramer's Rule Calculator" 於 https://MiniWebtool.com/zh-tw/克萊姆法則計算機/，來自 MiniWebtool，https://MiniWebtool.com/
by miniwebtool team. Updated: 2026-04-12
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