Gaussian Elimination Calculator – Solve Linear Systems
Solve linear equation systems using Gaussian elimination with step-by-step solutions
Table of Contents
How to Use
- Select the matrix size (2x2 or 3x3)
- Enter the coefficient matrix (one row per line)
- Enter the constant vector (space-separated)
- Click calculate to see the solution with steps
What is Gaussian Elimination?
Gaussian elimination is a method for solving systems of linear equations. It transforms the system's augmented matrix into row echelon form through elementary row operations, then uses back substitution to find the solution.
The method is named after German mathematician Carl Friedrich Gauss, though the technique was known to ancient Chinese mathematicians.
How Gaussian Elimination Works
- Form the augmented matrix [A|b] from the system Ax = b
- Forward elimination: Use row operations to create zeros below the diagonal
- The matrix is now in row echelon form (upper triangular)
- Back substitution: Solve for variables starting from the last equation
- Work upward to find all variable values
Elementary Row Operations
- Row switching: Swap two rows
- Row multiplication: Multiply a row by a non-zero constant
- Row addition: Add a multiple of one row to another row
Frequently Asked Questions
- When does a system have no solution?
- A system has no solution when the matrix is inconsistent, meaning the equations contradict each other. This shows up when row reduction produces a row like [0 0 | c] where c ≠ 0.
- What is a singular matrix?
- A singular matrix has a determinant of zero and represents a system with either no solution or infinitely many solutions. The calculator detects this and reports an error.
- Can this solve larger systems?
- This calculator handles 2×2 and 3×3 systems. For larger systems, specialized software is recommended due to numerical stability concerns.