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Cramer's Rule Calculator

Solve linear systems with Cramer's rule as a ratio of determinants, seeing each replaced matrix, its cofactor expansion and the exact fractional answer.

2×2
System size
-5
det(A)
unique solution
Status

Coefficient matrix A

23
1-1

det(A) = (2)(-1) − (3)(1) = -5

A1 — column 1 replaced by the constants

83
1-1

det(A1) = (8)(-1) − (3)(1) = -11

x = det(A1) / det(A) = -11 / -5 = 11/5 ≈ 2.2

A2 — column 2 replaced by the constants

28
11

det(A2) = (2)(1) − (8)(1) = -6

y = det(A2) / det(A) = -6 / -5 = 6/5 ≈ 1.2

Solution

x = 11/5 ≈ 2.2

y = 6/5 ≈ 1.2

Cramer’s rule gives each variable as a ratio of two determinants, so the answers stay exact fractions rather than rounded decimals.

What is the Cramer's Rule Calculator?

The ByteTools Cramer's Rule Calculator solves an n×n system of linear equations the way the method is actually taught: each unknown is the ratio xᵢ = det(Aᵢ) / det(A), where Aᵢ is the coefficient matrix with column i swapped for the constants.

  • Solves 2×2 up to 5×5 systems by the determinant-ratio method
  • Each replaced matrix Aᵢ shown with its swapped column highlighted
  • Cofactor expansion written out term by term for 2×2 and 3×3
  • Exact fractions plus a decimal approximation for every answer
  • Clear explanation when det(A) = 0 and the rule cannot be used
  • All computation is local to your browser

How to use the Cramer's Rule Calculator

  1. 1

    Enter your system as an augmented matrix, one equation per line — 2x + 3y = 8 becomes '2 3 8'.

  2. 2

    Read det(A) in the summary tiles; if it is zero, the rule does not apply and the tool says so.

  3. 3

    Look at each Aᵢ matrix, with the replaced column highlighted and its determinant shown.

  4. 4

    Read each variable as the ratio det(Aᵢ) / det(A), given as an exact fraction and a decimal.

  5. 5

    Copy the full solution with the copy button.

About the Cramer's Rule Calculator

The ByteTools Cramer's Rule Calculator solves an n×n system of linear equations the way the method is actually taught: each unknown is the ratio xᵢ = det(Aᵢ) / det(A), where Aᵢ is the coefficient matrix with column i swapped for the constants. It handles systems from 2×2 up to 5×5, and works in exact fractions so answers come out as 11/5 rather than 2.2000000001.

Every replaced matrix Aᵢ is printed with its swapped column highlighted, followed by its determinant. For 2×2 and 3×3 systems the cofactor expansion is written out term by term, so you can check the arithmetic against your own; for 4×4 and 5×5 the determinant is computed by exact elimination instead, because a full expansion would run to 24 or 120 terms.

When det(A) = 0 the tool says plainly that Cramer's rule does not apply, and explains that a zero determinant means the system has either no solution or infinitely many — a case you need Gauss-Jordan elimination to distinguish. Everything runs locally in your browser with no uploads.

Frequently asked questions

What is Cramer's rule?

It is a formula that gives each unknown in a square linear system as a ratio of two determinants: replace column i of the coefficient matrix with the constants, take that determinant, and divide by the determinant of the original matrix. It only works when the system has a unique solution.

When can you not use Cramer's rule?

Whenever det(A) = 0, since you would be dividing by zero. That happens when the equations are not independent, so the system has either no solution or infinitely many. Elimination will tell you which; Cramer's rule cannot.

Is Cramer's rule faster than elimination?

No — for anything beyond 3×3 it is dramatically slower, because it needs n+1 separate determinants. It is taught because it gives a clean closed-form expression and shows how the solution depends on the inputs, not because it is efficient.

How do you enter a system into this calculator?

Write each equation as its coefficients followed by the constant, one equation per line. The system 2x + 3y = 8 and x − y = 1 becomes two lines: '2 3 8' and '1 -1 1'. Missing variables need an explicit 0.

Why are the answers shown as fractions?

Because the ratio of two integer determinants is exactly a rational number. Showing 11/5 instead of 2.2 keeps the answer exact and matches what you get working it out by hand. A decimal approximation is shown alongside for convenience.

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