Cramer's Rule Calculator
Solve a 2- or 3-variable linear system with Cramer’s rule by entering the augmented coefficient matrix.
Computed via Cramer’s rule (Dx / D)
- 1
Main determinant D
2 × -3 − 1 × 1 = -7 - 2
Numerator determinant Dx
5 × -3 − 1 × -1 = -14Column 1 of the coefficient matrix is replaced with the constants. - 3
Solution x = Dx ÷ D
-14 ÷ -7 = 2
How does this calculator work?
Cramer’s rule solves a 2×2 or 3×3 linear system using determinants. Enter each equation as its coefficients plus the constant. The tool finds the main determinant D and each Dᵢ (column i swapped with the constants), then returns xᵢ = Dᵢ / D. If D = 0, no unique solution exists.
Formula
How this is calculated
Enter one equation per line in the augmented matrix box. Each line holds the coefficients of the variables followed by the constant on the right-hand side. Two lines define a 2×2 system (x, y); three lines define a 3×3 system (x, y, z). Values may be separated by spaces or commas.
The calculator splits the input into the square coefficient matrix A and the constant vector b. It computes the main determinant D = det(A). For each variable i it builds matrix Aᵢ by replacing column i of A with b, computes Dᵢ = det(Aᵢ), and then xᵢ = Dᵢ / D. The 2×2 determinant uses ad − bc; the 3×3 uses cofactor expansion along the first row.
Cramer’s rule only gives a unique solution when D ≠ 0. If D = 0 the system is either inconsistent (no solution) or dependent (infinitely many solutions), so the tool reports that no unique solution exists. Inputs must form a square system: each of the n rows needs exactly n + 1 numbers.
Frequently asked questions
If D = 0, Cramer’s rule cannot be applied because dividing by zero is undefined. The system has either no solution or infinitely many, and you should use Gaussian elimination to investigate further.
Write one equation per line as its coefficients followed by the constant. For 2x + y = 5 and x − 3y = −1, enter "2 1 5" and "1 -3 -1".
This calculator handles 2×2 and 3×3 systems. Cramer’s rule generalizes to larger ones, but it becomes computationally expensive, so elimination or matrix methods are preferred for big systems.
Also known as
TG we-Calculate Editorial Team. (2026). Cramer's Rule Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/cramers-rule-calculator
TG we-Calculate Editorial Team. "Cramer's Rule Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/cramers-rule-calculator.
TG we-Calculate Editorial Team, "Cramer's Rule Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/cramers-rule-calculator
@misc{wecalculate_cramers_rule_calculator, title = {Cramer's Rule Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/cramers-rule-calculator}}, year = {2026}, note = {TG we-Calculate} }
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