📊 Cramer's Rule Calculator
Solve system Ax = b using Cramer's rule: xᵢ = det(Aᵢ) / det(A)
How to Use This Calculator
Select System Size
Choose 2×2 or 3×3 for your system Ax = b.
Enter Matrix A and Vector b
Input the coefficient matrix A and right-hand side vector b.
Calculate
Click to solve using Cramer's rule: xᵢ = det(Aᵢ) / det(A).
Review Solution
See each xᵢ calculated from its corresponding determinant ratio.
Formula
xᵢ = det(Aᵢ) / det(A)
Where Aᵢ is matrix A with column i replaced by vector b
Cramer's Rule:
For system Ax = b, if det(A) ≠ 0:
x₁ = det([b | a₂ | a₃]) / det(A)
x₂ = det([a₁ | b | a₃]) / det(A)
x₃ = det([a₁ | a₂ | b]) / det(A)
where aᵢ are columns of A
Example: 2×2 System
a₁₁x₁ + a₁₂x₂ = b₁
a₂₁x₁ + a₂₂x₂ = b₂
x₁ = det([b₁ a₁₂; b₂ a₂₂]) / det(A)
x₂ = det([a₁₁ b₁; a₂₁ b₂]) / det(A)
Requirements:
- System must be square (n equations, n unknowns)
- det(A) ≠ 0 (A is invertible)
- Unique solution exists
About Cramer's Rule Calculator
Cramer's Rule Calculator solves a system of linear equations Ax = b using determinants. Each solution xᵢ is found by replacing column i of A with b, computing the determinant, and dividing by det(A). This method is elegant but computationally expensive for large systems.
When to Use This Calculator
- Small Systems: 2×2 or 3×3 systems
- Teaching: Understand determinant-based solutions
- Symbolic Computation: When formulas are needed
- Verification: Check solutions from other methods
Why Use Our Calculator?
- ✅ Step-by-Step: Shows each Aᵢ and its determinant
- ✅ Clear Formula: Displays xᵢ = det(Aᵢ) / det(A)
- ✅ Visual: Shows all intermediate matrices
- ✅ Educational: Helps understand Cramer's rule
- ✅ Accurate: Precise determinant calculations
- ✅ Free: No registration required
Key Concepts
- Cramer's Rule: xᵢ = det(Aᵢ) / det(A) where Aᵢ replaces column i with b
- Requirements: Square system with det(A) ≠ 0
- Efficiency: O(n!) for determinants, inefficient for large n
- Alternative: Gaussian elimination is O(n³) and more practical
- Advantage: Gives explicit formulas, useful for small systems
Limitations
Computational Cost: Requires computing n+1 determinants, which is O(n!) and impractical for large systems.
Singular Systems: If det(A) = 0, Cramer's rule cannot be applied. Use Gaussian elimination instead.
Frequently Asked Questions
What is Cramer's rule?
Cramer's rule solves Ax = b using determinants: xᵢ = det(Aᵢ) / det(A), where Aᵢ is A with column i replaced by b.
When can I use Cramer's rule?
Only when the system is square (n equations, n unknowns) and det(A) ≠ 0. If det(A) = 0, the system may have no solution or infinitely many solutions.
Why is Cramer's rule not used for large systems?
Computing determinants requires O(n!) operations, making it extremely slow. Gaussian elimination (O(n³)) is much more efficient for large systems.
What if det(A) = 0?
Cramer's rule cannot be applied. The matrix is singular (not invertible). The system may have no solution or infinitely many solutions - use Gaussian elimination to determine.
Is Cramer's rule faster than other methods?
No, it's slower due to determinant computations. However, it provides explicit formulas and is useful for small systems (2×2, 3×3) or symbolic computation.