Cramers Rule Solver Calculator
Cramer's rule gives a closed-form solution to a square linear system as ratios of determinants. For three unknowns you compute the determinant of the coefficient matrix, then three more determinants formed by swapping in the right-hand side column. Dividing each by the main determinant yields x, y, and z. This calculator takes three equations, each entered as the three coefficients plus the constant, and returns the solution along with the main determinant so you can see at once whether a unique solution exists.
Cramer's rule formula
D = det of the 3x3 coefficient matrix
Dx = D with column 1 replaced by constants
Dy = D with column 2 replaced by constants
Dz = D with column 3 replaced by constants
x = Dx/D, y = Dy/D, z = Dz/D (needs D != 0)
Each unknown is a ratio of two determinants. The numerator substitutes the right-hand side vector into the column for that unknown, and the denominator is the unchanged coefficient determinant.
Notes on Cramer's rule
- Each equation is four comma-separated numbers: three coefficients and a constant.
- A unique solution exists only when the main determinant D is nonzero.
- If D is zero the system has no unique solution and the tool returns n/a.
- The method is exact but inefficient for systems much larger than three unknowns.
- Substitute the solution back into each equation to verify.
Cramer's rule: frequently asked questions
What is Cramer's rule?
Cramer's rule solves a square system of linear equations by expressing each unknown as a ratio of determinants. The numerator replaces the coefficient column for that unknown with the right-hand side vector, and the denominator is the determinant of the coefficient matrix.
What is the formula for a 3x3 system?
Let D be the determinant of the coefficient matrix. Then x = Dx / D, y = Dy / D, z = Dz / D, where Dx, Dy, and Dz are the determinants formed by replacing the first, second, or third column with the right-hand side vector.
When does Cramer's rule fail?
Cramer's rule fails when the main determinant D is zero. A zero determinant means the system either has no solution or infinitely many, so no unique solution exists and the determinant ratios are undefined.
Is Cramer's rule efficient for large systems?
No. Cramer's rule is elegant for two or three unknowns but the number of determinant evaluations grows steeply with size. For larger systems Gaussian elimination is far more efficient and numerically stable.
How do I enter my equations?
Each row is one equation entered as four comma-separated numbers: the three coefficients followed by the right-hand side constant. For example 2, 1, -1, 8 represents 2x + y - z = 8.
Official sources
- NIST Digital Library of Mathematical Functions: Determinants and linear operators.
- NIST/SEMATECH e-Handbook of Statistical Methods: Linear algebra reference.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.