Cramer's Rule Calculator – Solve Linear Systems
Solve linear systems using Cramer's rule with determinants
Table of Contents
How to Use
- Select system size (2x2 or 3x3)
- Enter coefficient matrix rows on separate lines
- Enter constant vector values separated by spaces
- Click calculate to find the solution using Cramer's rule
What is Cramer's Rule?
Cramer's rule is a mathematical theorem used to solve systems of linear equations with the same number of equations as unknowns. It expresses the solution in terms of the determinants of matrices.
For a system Ax = b, where A is the coefficient matrix and b is the constant vector, each variable xᵢ is calculated as: xᵢ = det(Aᵢ) / det(A), where Aᵢ is the matrix A with its i-th column replaced by b.
When to Use Cramer's Rule
- Small systems (2x2 or 3x3) where manual calculation is feasible
- When you need the solution in terms of determinants
- Theoretical analysis and proofs
- Systems where the determinant is already known or easy to compute
For larger systems or numerical computation, methods like Gaussian elimination or LU decomposition are more efficient.
Limitations
- Only works when the coefficient matrix is square (same number of equations and unknowns)
- Requires det(A) ≠ 0 (matrix must be non-singular)
- Computationally inefficient for large systems (requires n+1 determinant calculations)
- Susceptible to numerical instability with ill-conditioned matrices
Example: 2x2 System
Solve: 2x + y = 8 and x + 3y = 13
det(A) = |2 1; 1 3| = 6 - 1 = 5
x = |8 1; 13 3| / 5 = (24 - 13) / 5 = 11/5 = 2.2
y = |2 8; 1 13| / 5 = (26 - 8) / 5 = 18/5 = 3.6
Frequently Asked Questions
- What happens if the determinant is zero?
- If det(A) = 0, the matrix is singular and the system either has no solution or infinitely many solutions. Cramer's rule cannot be used in this case.
- Is Cramer's rule efficient for large systems?
- No. Cramer's rule requires computing n+1 determinants for an n×n system, which becomes computationally expensive. Methods like Gaussian elimination are much more efficient for systems larger than 3×3.
- Can Cramer's rule solve systems with more equations than unknowns?
- No. Cramer's rule only applies to square systems (same number of equations as unknowns). For overdetermined or underdetermined systems, use least squares or other methods.