Cramer’s Rule Calculator – 2x2 & 3x3 Systems

Cramer’s rule solves square systems by ratios of determinants. For x + 2y = 5 and 3x − y = 1: the base determinant D = 1(−1) − 3(2) = −7. Replacing the x-column with the constants (5, 1) gives Dx = 5(−1) − 1(2) = −7, so x = Dx/D = 1; replacing the y-column gives Dy = 1(1) − 3(5) = −14, so y = 2. The calculator prints every 2×2 determinant expansion and extends to 3×3 systems with cofactor expansions shown. The zero-determinant case is handled explicitly: D = 0 means no unique solution (parallel or coincident constraints), and Cramer cannot proceed — elimination reveals whether solutions are none or infinite. The method shines for single-variable queries: computing just x requires only D and Dx, never the full reduction.

1. 1

   Enter the coefficient matrix and constants

   Square systems, 2×2 or 3×3

2. 2

   Compute the base determinant D

   The expansion is shown term by term

3. 3

   Build Dx, Dy, (Dz)

   Each variable column is replaced in turn

4. 4

   Divide

   x = Dx/D = 1, y = Dy/D = 2, with D = 0 flagged

• 1

  Dx swaps the constants into the x-column only

• 2

  D = 0 ⇒ Cramer inapplicable: no or infinite solutions

• 3

  Determinants scale: D halves when one equation is halved

• Replacing the wrong column for Dx

• Dividing D by Dx instead of Dx by D

• Forgetting to check D ≠ 0 before dividing

• Transcribing constants into the matrix in the wrong order