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Cramer's Rule Calculator

Solve systems of 2 or 3 linear equations using Cramer's rule. Enter coefficients, get step-by-step determinant calculations with animated matrix visualization, geometric interpretation graph, and the complete solution.

Cramer's Rule Calculator
Examples:
xy
[
a₁₁
a₁₂
a₂₁
a₂₂
] = [
b₁
b₂
]

Embed Cramer's Rule Calculator Widget

Cramer's Rule Calculator

The Cramer's Rule Calculator solves systems of 2 or 3 linear equations using determinants. Enter the coefficient matrix and constants vector, and get the complete solution with step-by-step determinant calculations, animated matrix visualization showing column replacement, and a geometric interpretation graph for 2×2 systems. Cramer's rule is a fundamental technique in linear algebra that expresses each variable as a ratio of two determinants.

What Is Cramer's Rule?

Cramer's rule is a theorem in linear algebra that provides an explicit formula for solving a system of linear equations with as many equations as unknowns, provided the system has a unique solution. Named after Swiss mathematician Gabriel Cramer (1704–1752), the rule uses determinants to express each variable as a ratio:

$$x_i = \frac{D_i}{D}$$

where \(D\) is the determinant of the coefficient matrix and \(D_i\) is the determinant formed by replacing the \(i\)-th column of the coefficient matrix with the constants vector.

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:

$$a_1x + b_1y = c_1$$ $$a_2x + b_2y = c_2$$

DeterminantFormulaDescription
\(D\)\(\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix} = a_1 b_2 - b_1 a_2\)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

The determinant of a 3×3 matrix is computed using cofactor expansion along the first row. Each \(D_i\) is formed by replacing the corresponding column with the constants vector, and the solution is \(x_i = D_i / D\).

When Does Cramer's Rule Work?

ConditionD ValueResult
Unique solutionD ≠ 0Each variable = D_i / D
No solution (inconsistent)D = 0, some D_i ≠ 0Lines/planes are parallel
Infinitely many solutionsD = 0, all D_i = 0Equations are dependent

Cramer's Rule vs. Other Methods

MethodBest ForLimitation
Cramer's RuleSmall systems (2×2, 3×3), exact symbolic solutionsSlow for large systems (n! complexity)
Gaussian EliminationGeneral systems, large matricesNo closed-form formula
Matrix InverseMultiple right-hand sidesRequires D ≠ 0, expensive to compute
LU DecompositionRepeated solving, numerical stabilityMore complex to implement

How to Use the Cramer's Rule Calculator

  1. Choose the system size: Select 2×2 or 3×3 depending on how many equations and unknowns you have.
  2. Enter coefficients: Fill in the coefficient matrix on the left. Each row corresponds to one equation, and each column to a variable (x, y, z).
  3. Enter constants: Fill in the constants vector on the right (the right-hand side of each equation).
  4. Click Solve: The calculator computes all determinants (D, D_x, D_y, and optionally D_z), determines the solution type, and shows the step-by-step process with animated matrix visualization.

Real-World Applications

FieldApplicationExample
EngineeringCircuit analysis (Kirchhoff's laws)Finding currents in a resistor network
EconomicsMarket equilibriumSupply and demand intersection
PhysicsForce balanceFinding reaction forces in statics
ChemistryBalancing equationsStoichiometric coefficients
Computer GraphicsCoordinate transformationsLine/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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