Cramer's Rule Calculator
Solve systems of linear equations using determinants and matrix algebra
System of Linear Equations
First equation
0 = 0
Second equation
0 = 0
Solution
Infinite Solutions
The system has infinitely many solutions. All determinants are zero, indicating dependent equations.
Example Problems
2×2 System Example
2x + 3y = 7
x - y = 1
Solution: x = 2, y = 1
3×3 System Example
x + y + z = 6
2x - y + z = 3
x + 2y - z = 2
Solution: x = 1, y = 2, z = 3
Determinant Formulas
2×2 Matrix
det([a b; c d]) = ad - bc
3×3 Matrix
det([a b c; d e f; g h i]) =
aei + bfg + cdh - ceg - bdi - afh
Quick Tips
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If det(W) = 0, system has no unique solution
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Cramer's rule only works for square systems
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Each variable matrix replaces one column with constants
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Solution is determinant ratio: x = det(Wₓ)/det(W)
Understanding Cramer's Rule
What is Cramer's Rule?
Cramer's rule is a mathematical theorem that provides an explicit formula for solving systems of linear equations using determinants. It's named after Gabriel Cramer, who published the rule in 1750, though the method was known earlier.
When to Use Cramer's Rule
- •System has the same number of equations and unknowns
- •Coefficient matrix has non-zero determinant
- •Need exact solutions rather than approximations
- •Working with small systems (2×2 or 3×3)
Method Overview
- 1. Form the coefficient matrix W from the system
- 2. Calculate det(W) - if zero, no unique solution exists
- 3. For each variable, create a new matrix by replacing the corresponding column with constants
- 4. Calculate determinants of these new matrices
- 5. Divide each variable determinant by det(W)
Advantages & Limitations
Advantages
- • Direct formula method
- • Exact solutions
- • No elimination steps
- • Clear systematic approach
Limitations
- • Only for square systems
- • Computationally expensive for large systems
- • Requires non-zero determinant
- • Numerical instability possible