Gauss-Jordan Elimination Calculator

$\color{red} \bold{Related \space Calculators:}$

Inverse of a matrix
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Gaussian Elimination
Solving a system of linear equation
Characteristic polynomial
Eigen values and Eigen vectors
Nature of solution for a system of linear equation.

$\bold{Table \space of \space Content}$

• 1. Introduction to the Gauss-Jordan Elimination
• 2. What is the Formula used & conditions required ?
• 3. How do I apply the Gauss-Jordan Elimination to a system of linear equations?
• 4. Why choose the Gauss-Jordan Elimination Calculator?
• 5. A Video for explaining this concept
• 6. How to use this calculator ?
• 7. Solved Examples
• 8. Frequently Asked Questions (FAQs)
• 9. What are the real-life applications?
• 10. Conclusion

1. Introduction to the Gauss-Jordan Elimination

Have you ever wished for a streamlined method to solve systems of linear equations without getting lost in complex calculations? Gauss-Jordan Elimination is your key to simplifying the process. In this blog, we'll journey through Gauss-Jordan Elimination, uncovering its importance, practical applications, and knack for making linear algebra accessible.
$\bold{Definition}$
Gauss-Jordan Elimination is an advanced method for solving systems of linear equations by transforming an augmented matrix into its reduced row echelon form. It's a refined version of Gaussian Elimination, taking the process one step further to achieve the most simplified and organized matrix structure.

2. What is the Formula used & conditions required?

$\bold{Formula \space used}$
The formula for Gauss-Jordan Elimination builds upon Gaussian Elimination, involving elementary row operations like swapping rows, multiplying a row by a constant, and adding or subtracting multiples of one row from another.
$\bold{Conditions \space required}$
Conditions include consistent dimensions and avoiding division by zero.

3. How do I calculate the Gauss-Jordan Elimination of a matrix?

Write the augmented matrix for the given system of linear equations.
Apply necessary row transformations to reduce the augmented matrix in reduced row echelon form.
Use the back-substitution method to find the value of each variable.

4. Why choose our Gauss-Jordan Elimination Calculator?

$\bold{Easy \space \space to \space Use}$
Our calculator page provides a user-friendly interface that makes it accessible to both students and professionals. You can quickly input your square matrix and obtain the matrix of minors within a fraction of a second.

$\bold{Time \space Saving \space By \space automation}$
Our calculator saves you valuable time and effort. You no longer need to manually calculate each cofactor, making complex matrix operations more efficient.

$\bold{Accuracy \space and \space Precision}$
Our calculator ensures accurate results by performing calculations based on established mathematical formulas and algorithms. It eliminates the possibility of human error associated with manual calculations.

$\bold{Versatility}$
Our calculator can handle all input values like integers, fractions, or any real number.

$\bold{Complementary \space Resources}$
Alongside this calculator, our website offers additional calculators related to Pre-algebra, Algebra, Precalculus, Calculus, Coordinate geometry, Linear algebra, Chemistry, Physics, and various algebraic operations. These calculators can further enhance your understanding and proficiency.

5. A video based on how to apply the Gauss-Jordan Elimination method to a system of linear equations.

6. How to use this calculator

This calculator will help you to solve the system of linear equations by the Gauss-Jordan elimination method.
In the given input boxes, you have to put the value of the augmented matrix.
A step-by-step solution will be displayed on the screen after clicking the Calculate button.
You can access, download, and share the solution.

7. Solved Example

$\bold{Question}$
Apply Gauss-Jordan Elimination to solve for x,y, and z for the following system of linear equations
2x+3y−z=8
4x−y+2z=6
3x+2y−3z=1
$\bold{Solution}$
$\bold{Step \space 1:}$ Write the augmented matrix = $\begin{bmatrix} 2 & 3 & -1 & 8 \\ 4 & -1 & 2 & 6 \\ 3 & 2 & -3 & 1 \end{bmatrix}$
$\bold{Step \space 2:}$ Convert the above matrix to Reduced Row echelon form = $\begin{bmatrix} 1 & 3 & 0 & \frac{39}{41} \\ 0 & 1 & 0 & \frac{118}{41} \\ 0 & 0 & 1 & \frac{104}{41} \end{bmatrix}$
$\bold{Step \space 3:}$ Use the back-substitution method to solve for x, y and z.

8. Frequently Asked Questions (FAQs)

What distinguishes Gauss-Jordan Elimination from Gaussian Elimination?

Gauss-Jordan Elimination takes the process further, ensuring each column containing a leading entry has only zeros above and below it, resulting in the most simplified form.

Can Gauss-Jordan Elimination handle systems with infinite solutions?

Yes, Gauss-Jordan Elimination effectively identifies systems with infinite solutions and clearly represents them.

Why use Gauss-Jordan Elimination instead of other methods?

Gauss-Jordan Elimination is preferred for its efficiency in obtaining the most simplified and organized form, making it a popular choice for solving linear systems.

Can it be applied to non-square matrices?

Yes, Gauss-Jordan Elimination applies to both square and non-square matrices, providing a systematic solution to systems of linear equations.

Is there a risk of division by zero during Gauss-Jordan Elimination?

While division by zero should be avoided, the method is designed to handle such situations and identify when a solution is impossible.

9. What are the real-life applications?

Gauss-Jordan Elimination finds practical use in various fields, including engineering, physics, and computer science. Its efficiency in solving systems of linear equations contributes to optimizing complex systems and simulations.

10. Conclusion

Gauss-Jordan Elimination emerges as a powerful tool, offering a refined approach to solving linear systems. As you navigate its applications, remember that Gauss-Jordan Elimination is more than just a mathematical technique – it's a problem-solving strategy that transforms complex scenarios into streamlined solutions. Embrace its efficiency and witness how Gauss-Jordan Elimination simplifies the landscape of linear algebra, making it accessible and applicable in diverse domains.

This blog is written by Neetesh Kumar

If you have any suggestions regarding the improvement of the content of this page, please write to me at My Official Email Address: [email protected]

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