The Gaussian Elimination Calculator is a powerful mathematical tool used to solve systems of linear equations using a method known as Gaussian elimination. This process involves three types of elementary row operations that simplify the matrix into a form that can be easily interpreted to find the solutions. These solutions are pivotal in fields ranging from engineering to economics, making the calculator essential for students, professionals, and researchers.
Elementary Row Operations
- Row Switching (Interchange): Swap two rows of the matrix.
- Row Scaling (Multiplication): Multiply all elements of a row by a non-zero scalar.
- Row Addition (or Subtraction): Add (or subtract) a scalar multiple of one row to another row.
Gaussian Elimination Algorithm
- Forward Elimination: Convert the matrix into row-echelon form.
- Back Substitution: Solve the resulting system of equations by substitution.
Formulas of Gaussian Elimination Calculator
Let’s denote the augmented matrix as [A | b], where A is the coefficient matrix, and b is the column vector of constants.
- Row Switching: To interchange rows i and j, perform Ri <-> Rj.
- Row Scaling: To multiply row i by a scalar k, perform Ri -> k * Ri.
- Row Addition (or Subtraction): To add (or subtract) a multiple k of row j to (from) row i, perform Ri -> Ri + k * Rj.
Conversion Table for Common Terms
The following table includes general terms and their definitions to aid in understanding and utilizing the Gaussian Elimination Calculator effectively:
| Term | Definition |
|---|---|
| Linear System | A collection of linear equations involving the same set of variables. |
| Coefficient Matrix | A matrix consisting of coefficients of the variables in the linear equations. |
| Augmented Matrix | A matrix obtained by appending the constants of the equations as an additional column to the coefficient matrix. |
| Row-Echelon Form | A form of matrix where all leading entries are 1s, and all elements below these leading entries are 0s. |
| Leading Entry | The first non-zero element in a row, moving from left to right. |
| Back Substitution | A method used to find the solution of a system once the matrix is in row-echelon form. |
Example of Gaussian Elimination Calculator
Problem: Solve the system of equations given by:
3x + 4y – z = 5
2x – 2y + 4z = -2
-x + 0.5y – z = 0
Solution Using Gaussian Elimination:
- Construct the Augmented Matrix:
| 3 4 -1 | 5 |
| 2 -2 4 | -2 |
| -1 0.5 -1 | 0 |
- Apply Row Operations to Achieve Row-Echelon Form:
R2 -> R2 - (2/3)R1R3 -> R3 + (1/3)R1
- Further Simplification to Reach Row-Echelon Form:
- Continue applying row operations until the left part of the augmented matrix (coefficient matrix) forms an upper triangular matrix.
- Back Substitution to Find the Solutions:
- Simplify the upper triangular matrix and solve for the variables using back substitution.
Resulting System in Row-Echelon Form:
| 3 4 -1 | 5 |
| 0 -4.67 5.33 | -4.33 |
| 0 0 -0.33 | 0.33 |
Final Solution Set:
x = 3, y = 2, z = -1
Most Common FAQs
Gaussian Elimination is used to solve systems of linear equations. It transforms the system into a triangular matrix form using elementary row operations, making the variables easier to solve for systematically.
The Gaussian Elimination Calculator is highly accurate provided the input data is correct and the equations do not form a singular (non-invertible) matrix, ensuring that the system has a unique solution or is consistent.
Yes, Gaussian Elimination can handle matrices of any size, including non-square matrices. However, for non-square matrices, the system might be underdetermined (more variables than equations) or overdetermined (more equations than variables), and special cases like these require additional steps or adjustments in the elimination process.
