Cramer’s rule gives an explicit formula for the solution of a square linear system. It is elegant for small systems and for theory, though elimination is faster for large ones.
How to use the Cramer’s Rule Calculator
- Number of equations
- a₁ (x)
- b₁ (y)
- c₁ (z)
- d₁ (right side)
- a₂ (x)
- b₂ (y)
- c₂ (z)
- d₂ (right side)
- a₃ (x)
- b₃ (y)
- c₃ (z)
- d₃ (right side)
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
The determinant of the coefficient matrix W is computed, then the determinants with each column replaced by the constants. If |W| = 0 the system has no unique solution.
Formula
x = |Wₓ| / |W|, y = |W_y| / |W|, z = |W_z| / |W|
Wₓ: W with the x-column replaced by the right-hand sides
Worked example
Inputs — Number of equations: Two equations (x, y); a₁ (x): 2; b₁ (y): 3; d₁ (right side): 12; a₂ (x): 3; b₂ (y): -1; d₂ (right side): 7.
Results — Solution = x = |Wx| / |W| = −33 / −11 = 3
y = |Wy| / |W| = −22 / −11 = 2; |W| = −11; |Wₓ| = −33; |W_y| = −22; x = 3; y = 2.
Key terms
- Determinant
- A scalar computed from a square matrix.
Frequently asked questions
When can I use Cramer's rule?
When the coefficient determinant is non-zero.
What does
W| = 0 mean? | No solution or infinitely many – Cramer's rule cannot decide which on its own.
Is Cramer's rule efficient?
For 2 × 2 and 3 × 3 systems yes; for larger systems use elimination.
Who was Cramer?
Gabriel Cramer, a Swiss mathematician who published the rule in 1750.