Cramer’s Rule Calculator (2×2 and 3×3 Systems, Free)

A Cramer’s rule calculator solves a square linear system with determinants: each unknown equals D_x/D, where D is the coefficient determinant and D_x replaces the x-column with the constants. For 2x + 3y = 8 and x − y = −1, D = −5, D_x = −5 and D_y = −10, so x = 1 and y = 2.

How to use the Cramer’s rule calculator

1. Pick the size: a 2 × 2 system (x, y), a 3 × 3 system (x, y, z) or a 4 × 4 system (w, x, y, z).
2. Enter each equation’s coefficients and constant in the grid, writing 0 for a missing variable. Whole numbers, decimals and fractions such as −3/4 all work. You can also type the equations, for example 2x + 3y = 8; x − y = −1, and press Fill grid.
3. Press Solve (the answer also updates as you type). You get D, every replaced-column determinant, the solution as exact fractions with decimals, the graph of the two lines for a 2 × 2 system, and full steps with a check.

How Cramer’s rule works

Write the system as AX = B, where A is the square matrix of coefficients and B is the column of constants. Cramer’s rule says that when D = det(A) ≠ 0, each variable is

x_i = D_i / D, where D_i is the determinant of A with column i replaced by B.

For a 2 × 2 system a₁x + b₁y = c₁ and a₂x + b₂y = c₂:

• D = a₁b₂ − b₁a₂
• D_x = c₁b₂ − b₁c₂ (the x-column replaced by the constants)
• D_y = a₁c₂ − c₁a₂ (the y-column replaced by the constants)
• x = D_x/D and y = D_y/D

For a 3 × 3 system you need four determinants (D, D_x, D_y, D_z), and for 4 × 4 five. The calculator expands each 3 × 3 determinant along the first row, a(ei − fh) − b(di − fg) + c(dh − eg), and each 4 × 4 determinant into four 3 × 3 minors, using exact fraction arithmetic so no rounding creeps in.

Worked examples

2 × 2 system

Solve 2x + 3y = 8 and x − y = −1. D = 2(−1) − 3(1) = −5. D_x = 8(−1) − 3(−1) = −5 and D_y = 2(−1) − 8(1) = −10. So x = −5/−5 = 1 and y = −10/−5 = 2. Check: 2(1) + 3(2) = 8 and 1 − 2 = −1.

2 × 2 system with a fraction answer

For 3x + 2y = 7 and 4x − 5y = 3: D = 3(−5) − 2(4) = −23, D_x = 7(−5) − 2(3) = −41 and D_y = 3(3) − 7(4) = −19. So x = 41/23 ≈ 1.7826 and y = 19/23 ≈ 0.8261. Cramer’s rule gives these exact fractions directly.

3 × 3 system

Solve 2x + y − z = 8, −3x − y + 2z = −11 and −2x + y + 2z = −3. Expanding along the first row, D = 2(−4) − 1(−2) + (−1)(−5) = −1. Replacing one column at a time gives D_x = −2, D_y = −3 and D_z = 1, so x = 2, y = 3 and z = −1.

When D = 0

For 2x + 4y = 6 and x + 2y = 5, D = 2(2) − 4(1) = 0, so Cramer’s rule cannot divide. Here D_x = 6(2) − 4(5) = −8 ≠ 0, which for a 2 × 2 system means the lines are parallel and there is no solution. If D, D_x and D_y were all 0 (as for x + 2y = 3 and 2x + 4y = 6), the equations would describe the same line and there would be infinitely many solutions.

No unique solution: what D = 0 tells you

A zero determinant means the coefficient rows are linearly dependent, so the system has either no solution or infinitely many. Cramer’s rule alone cannot always tell which. For 2 × 2 systems the replaced determinants decide it, as in the example above. For 3 × 3 and larger systems, D = D_x = D_y = D_z = 0 can still happen when the system has no solution (for example x + y + z = 1, x + y + z = 2, x + y + z = 3). That is why the calculator row reduces the augmented matrix [A | B] whenever D = 0: if a row becomes 0 = c with c ≠ 0 there is no solution; otherwise it writes the infinite family in terms of free variables, such as x = 2 − (1/2)z, y = 1 − (1/2)z, z any real number.

Cramer’s rule or elimination?

Common mistakes

Checking homework with the calculator

Solve the problem by hand first: compute D, then each replaced determinant, then divide. Enter the same system here and compare your D and each D_i with the calculator’s values one at a time; the first one that differs shows exactly where the slip happened. The last step substitutes the answer back into every equation, a habit worth copying on tests.

Cramer’s rule practice quiz: 10 questions

Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.

Frequently asked questions

How do you solve a system with Cramer’s rule?

Find the coefficient determinant D. For each variable, replace that variable’s column with the constants and find the determinant, then divide it by D. For 2x + 3y = 8 and x − y = −1, D = −5, D_x = −5 and D_y = −10, so x = 1 and y = 2.

Does Cramer’s rule always work?

Only when the system is square and D ≠ 0. If D = 0 the system has no solution or infinitely many, and you need elimination to tell which. For example, x + y = 2 and 2x + 2y = 4 has D = 0 and infinitely many solutions.

How do you do Cramer’s rule on a TI-84 calculator?

Enter the coefficient matrix as [A] in the matrix editor (2nd, x⁻¹) and use MATH, 1:det( to find D. Then edit [A] so that one column holds the constants (or store a copy as [B]), find its determinant, and divide by D. Repeat for each variable.

How do you use Cramer’s rule for a 3 × 3 system?

Compute four 3 × 3 determinants: D, D_x, D_y and D_z, each by expanding along a row. Then x = D_x/D, y = D_y/D and z = D_z/D. For a triangular system such as x + y + z = 6, y + z = 5, z = 3, D is the product of the diagonal, 1, and the solution is x = 1, y = 2, z = 3.

Can Cramer’s rule be used with complex numbers?

Yes. The determinant formulas work for complex coefficients exactly as for real ones, as long as D ≠ 0. This calculator handles real numbers (whole numbers, decimals and fractions).

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