Cramer's Rule Calculator
Solve systems of linear equations using determinants and Cramer's rule.
System of Equations
Coefficient Matrix A | Constants b
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System Representation
2x1 + 1x2 - 1x3 = 8
3x1 - 1x2 + 2x3 = -11
2x1 + 1x2 + 2x3 = -3
Solution
x1 = 2.000000
x2 = 3.000000
x3 = -1.000000
det(A) = -1.000000
Determinants for Each Variable
det(A1) = -2.0000 → x1 = -2.0000 / -1.0000 = 2.0000
det(A2) = -3.0000 → x2 = -3.0000 / -1.0000 = 3.0000
det(A3) = 1.0000 → x3 = 1.0000 / -1.0000 = -1.0000
Verification (Ax = b)
Row 1: 8.0000 = 8.0000
Row 2: -11.0000 = -11.0000
Row 3: -3.0000 = -3.0000
Cramer's Rule
Formula
xᵢ = det(Aᵢ) / det(A)
Where Aᵢ is A with column i replaced by b
Requirements
- Square coefficient matrix
- det(A) ≠ 0 (unique solution exists)