Cramers Rule Calculator - 3x3 Systems by Determinant

A Cramer's rule calculator solves a square linear system A x = b by turning it into four scalar determinants: the main determinant D of A and the three Cramer determinants Dx, Dy, Dz from replacing each column of A with the constants vector b. Type the nine entries of A and the three entries of b, and the page returns the four determinants and the unique solution x = Dx / D, y = Dy / D, z = Dz / D.

• • Linear-algebra homework and 3x3 system drills: Confirm a problem that asks for the unique solution of a 3x3 system A x = b and show the four determinants that justify the answer.
• • Cross-checking matrix-inverse and elimination answers: Solve the same system with Cramer's rule to check a result from row reduction, the matrix inverse, or a numerical solver.
• • Detecting singular matrices and rank problems: Use the main determinant D to spot a singular coefficient matrix in a 3x3 system before a long elimination.

Cramer's rule is the determinant-based cousin of the matrix-inverse approach: where the inverse method writes x = A⁻¹ b, Cramer's rule replaces the inverse step with three small determinants that you can compute by hand or with the rule of Sarrus.

To solve the same 3x3 system with x = A⁻¹ b, the Matrix Inverse Calculator page runs the matrix inverse on the same nine entries and returns the solution from a single inverse step.

The page takes the twelve inputs, builds the main determinant D and the three Cramer determinants Dx, Dy, Dz, and divides each Cramer determinant by D whenever D is non-zero. If D comes out as zero, the system has no unique solution and the page returns a singular-matrix message.

• a11, a12, a13: First row of the 3x3 coefficient matrix A.
• a21, a22, a23: Second row of A.
• a31, a32, a33: Third row of A.
• b1, b2, b3: Right-hand-side constants of the three equations.
• D = det(A): Main determinant of A, computed by the rule of Sarrus or a cofactor expansion.
• Dx, Dy, Dz: Determinants of A with column 1, 2, or 3 replaced by the column vector b.

The page uses the standard cofactor expansion along the first row for every determinant, so D, Dx, Dy, and Dz share the same arithmetic. Swap any two rows of the augmented matrix [A | b] and D and every Cramer determinant flip sign, so each ratio Di / D and the solution x, y, z stays the same; this matches the row-swap sign rule the existing determinant calculator shows on its worked example card.

According to Encyclopaedia Britannica, Cramer's rule solves A x = b by computing the main determinant D of A and three Cramer determinants Dx, Dy, Dz, then setting x = Dx / D, y = Dy / D, and z = Dz / D whenever D is non-zero.

When the same 3x3 matrix shows up elsewhere and you only need the scalar det(A), the Determinant Calculator page returns the determinant with the Sarrus pattern and the Laplace cofactor expansion.

Four ideas explain why Cramer's rule works and when it returns a clean answer or fails.

The cofactor expansion the calculator uses for each 3x3 determinant is the same expansion the cofactor matrix page shows row by row, so a cofactor breakdown is one click away.

If a textbook asks for a cofactor-by-cofactor breakdown of the main determinant or a Cramer determinant, the Cofactor Matrix Calculator page shows the cofactor expansion row by row on the same matrix.

Five short steps cover every common 3x3 case, from a clean textbook example to a singular matrix.

1. 1 Enter the first row of A: Type a11, a12, and a13. The default is 2, 1, 3, the first row of the worked example.
2. 2 Enter the second and third rows of A: Fill in a21, a22, a23 and then a31, a32, a33. Read the matrix row by row, then check the layout before moving to the constants.
3. 3 Enter the constants b1, b2, b3: Type the right-hand side of the three equations. The default 6, 4, 6 are the worked-example constants, so the page starts as a solved Cramer's rule problem with x = y = z = 1.
4. 4 Read x, y, z and the four determinants: The primary outputs are x, y, and z. The page also returns D = det(A) and the three Cramer determinants Dx, Dy, Dz so you can verify the xi = Di / D step.
5. 5 Reset or paste in a new system: Click Reset to return to the worked example. If D = 0, the page shows a singular-matrix message instead of x, y, z.

Try the system 3x + 2y = 7, x + 4y = 9, z = 0: set a33 = 1 and a31, a32, b3 = 0. The page reports D = 10, Dx = 10, Dy = 20, Dz = 0, and the 2x2 reduction x = 1, y = 2, z = 0.

Once you have x, y, z from the Cramer's rule page, the Matrix Multiplication Calculator tool can sanity-check the answer by multiplying A back into the vector and confirming that the product equals b.

These benefits matter most when you want a quick, transparent check of a 3x3 system without paper.

• • Skip the four-determinant arithmetic: A 3x3 Cramer's rule problem asks for four scalar determinants and three divisions. The calculator handles all four with the same cofactor expansion, so you can focus on setting up the system.
• • See the full xi = Di / D breakdown: The page shows D, Dx, Dy, Dz, and the final x, y, z in one place, which makes it a good way to check your own Cramer's rule work.
• • Catch singular matrices before they bite you: When two rows of A are linearly dependent, D = 0 and the page returns a singular-matrix message instead of a misleading finite x, y, z.
• • Cross-check a matrix-inverse or row-reduction answer: If you solved the same system with the matrix inverse or Gaussian elimination, the Cramer's rule numbers should match, and the matrix-rank calculator can confirm the rank that D = 0 implies.

Use the page as a check, not as a replacement for understanding the method.

If the page shows a singular-matrix message because D = 0, the Matrix Rank Calculator page confirms the rank that D = 0 implies and tells you whether the system has no solution or infinitely many.

Cramer's rule is the same formula in every case, but a few factors change what the result means.

• • This page is the 3x3 Cramer's rule case only. For a 2x2 system, use the standard ad - bc shortcut or pair this page with the existing determinant calculator.
• • Cramer's rule requires D to be non-zero. When D = 0, the rule returns no unique x, y, z, and the matrix inverse does not exist either; use row reduction (Gaussian elimination) or rank analysis on the augmented matrix [A | b] to characterize the solutions.

According to Wolfram MathWorld, Cramer's rule expresses the solution of a square linear system A x = b as a ratio of determinants, with xi = det(A_i) / det(A), and requires det(A) to be non-zero for a unique solution.

To express one row of A as a linear combination of the other two to explain why D = 0, the Linear Combination Calculator page solves the matching system on the same 3x3 input.