Cramer's Rule Calculator

Enter the coefficients of a 2×2, 3×3, or 4×4 system of linear equations and solve using Cramer's Rule. Every determinant is computed and displayed step by step so you can follow the full working.

Number of variables:

x₁	x₂	x₃	= b

Notes

How Cramer's Rule Works

For a system Ax = b, Cramer's Rule computes each variable xᵢ as the ratio of two determinants: the determinant of A with its i-th column replaced by b, divided by det(A).

x_{i} = \frac{D _{i}}{D}, D = det (A), D_{i} = det (A_{i})

ℹIf D = 0 and all Dᵢ = 0, the system has infinitely many solutions. If D = 0 but any Dᵢ ≠ 0, the system has no solution.

Worked Example — 2×2

System: 2x₁ + x₂ = 5, x₁ − x₂ = 1

D = 21 1 - 1 = (2) (- 1) - (1) (1) = - 3

D_{1} = 51 1 - 1 = - 6, D_{2} = 2151 = - 3

x_{1} = \frac{- 6}{- 3} = 2, x_{2} = \frac{- 3}{- 3} = 1

Worked Example — 3×3

System: x₁ + x₂ + x₃ = 6, 2x₁ − x₂ + x₃ = 3, x₁ + 2x₂ − x₃ = 2

D = 121 1 - 1 2 11 - 1 = 1 (1 - 2) - 1 (- 2 - 1) + 1 (4 + 1) = - 1 + 3 + 5 = 7

D_{1} = 632 1 - 1 2 11 - 1 = 7, D_{2} = 121632 11 - 1 = 7, D_{3} = 121 1 - 1 2 632 = 7

x_{1} = \frac{7}{7} = 1, x_{2} = \frac{7}{7} = 1, x_{3} = \frac{7}{7} = 1

ℹVerify: 1+1+1 = 6 ✓, 2(1)−1+1 = 2 ✓, 1+2(1)−1 = 2 ✓

Cramer's Rule — Full Notes — Theory, derivation, and multiple worked examples
Cramer's Rule Formula Reference — Formula, variables, and step-by-step usage

Frequently Asked Questions

What is Cramer's Rule?

Cramer's Rule is a method for solving a system of n linear equations with n unknowns using determinants. Each variable xᵢ equals Dᵢ/D, where D is the determinant of the coefficient matrix and Dᵢ is the determinant with the i-th column replaced by the constants.

When does Cramer's Rule fail?

Cramer's Rule fails when D = det(A) = 0, meaning the coefficient matrix is singular. If D = 0 and all Dᵢ = 0, the system has infinitely many solutions. If D = 0 and any Dᵢ ≠ 0, the system is inconsistent and has no solution.

Is Cramer's Rule efficient for large systems?

No. Computing determinants has O(n!) complexity by cofactor expansion. For large systems (n > 4), Gaussian elimination or LU decomposition is far more efficient. Cramer's Rule is mainly used for 2×2 and 3×3 systems in education.

Can Cramer's Rule handle non-square systems?

No. Cramer's Rule requires the coefficient matrix to be square (n×n). For non-square systems, use Gaussian elimination or least squares methods.