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Gaussian Elimination Calculator

Gaussian Elimination Calculator

Set the matrix of a linear equation and write down entries of it to determine the solution by applying the gaussian elimination method by using this calculator.


Size of the matrix:




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This free gaussian elimination calculator matrix is specifically designed to help you in resolving systems of equations. Yes, now getting the most accurate solution of equations is just a couple of clicks away.

Let’s move on and understand the concept of this algorithm to find the solution of matrix equations.

Stay focused!

What Is Gaussian Elimination Algorithm?

In the light of mathematical analysis:

“The particular method that is used to find solution to the linear equations by arranging the augmented matrix of their coefficient numbers is known as the Gaussian Algorithm”

Augmented Matrix:

“An augmented matrix is a special matrix that consists of all the constants of the linear equations.”

Generic Form:

$$ A = \begin{bmatrix} a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3 \\\end{bmatrix} \hspace{0.25in} B = \begin{bmatrix}d_1\\d_2\\d_3\\\end{bmatrix} $$

$$ \text{Augmented Matrix} = \left[\begin{array}{ccc|c}a_1&b_1&c_1&d_1\\a_2&b_2&c_2&d_2\\a_3&b_3&c_3&d_3\\\end{array}\right] $$

The best augmented matrices calculator also displays the final answer in the same format as aforementioned above.

If we write the system of linear equations using the coefficients of the augmented matrix, then we get:

$$ a_{1}x + b_{1}y + c_{1}z = d_{1} $$

$$ a_{2}x + b_{2}y + c_{2}z = d_{2} $$

$$ a_{3}x + b_{3}y + c_{3}z = d_{3} $$

The final answer that is also determined by the free gaussian elimination calculator is represented in the following format:

$$ \left[\begin{array}{ccc|c}1&0&0&k\\0&1&0&l\\0&0&1&m\\\end{array}\right] $$

Properties of Augmented Matrix:

An augmented matrix possesses the following properties:
So here we have the following format of the ,atrix that is finalised as the under form of the matrix:

  • The rectangular augmented matrix
  • The variables in the linear equations and the constant term determine the number of columns.
  • The number of systems of equations is the same as the number of rows.
  • The augmented matrix’s rows can be swapped around.
  • A constant can be used to multiply or divide the elements of a certain row.
  • The specific row of the matrix can be added to and removed from other rows.
  • A matrix row’s multiple can be applied to another matrix row.

How to Apply Gaussian Elimination Algorithm?

Here we are going to apply this theorem on an example below. So for better understanding, just stay focused!

Example # 01:

Find solution of the following system of equations as under:

$$ 3x_{1} + 6x_{2} = 23 $$

$$ 6x_{1} + 2x_{2} = 34 $$


No doubt our widely used gaussian elimination calculator with steps will show detailed calculations to simplify these equations, but we need to analyse the scenario manually.

The equivalent augmented matrix form of the above equations are as follows:

$$ \begin{bmatrix} 3&6&23 \\ 6&2&34 \\\end{bmatrix} $$

Gaussian Elimination Steps:

Step # 01:

Divide the zeroth row by 3.

$$ \left[\begin{array}{cc|c}1&2& \frac{23}{3} \\6&2&34 \\\end{array}\right] $$

Step # 02:

Multiply the first row by 6 and then subtract it from the zeroth row.

$$ \left[\begin{array}{cc|c}1&2&\frac{23}{3} \\0&-10&-12 \\\end{array}\right] $$

Step # 03:

Go for dividing the first row by -10.

$$ \left[\begin{array}{cc|c}1&2&\frac{23}{3} \\0&1&\frac{6}{5}\\\end{array}\right] $$

Step # 04:

Get going for finding the product of zeroth row and 2. After doing that, subtract the result from the first row.

$$ \left[\begin{array}{cc|c}1&0&\frac{26333333334}{5000000000}\\0&1& \frac{6}{5}\\\end{array}\right] $$

As you see on the left side of the matrix, we get the identity matrix. So the answer on the right side pof the equation would be the values of the variables in the equations.

So the final results are as follows:

$$ b_{1} = 5.266 $$

$$ b_{2} = 1.2 $$

The same results can also be verified by using outer free gauss-jordan elimination calculator.

How Gaussian Elimination Calculator Works?

Get going to understand how this free gaussian elimination solver matrix row reduction algorithm simplifies equation systems.


  • First, set up the order of the matrix from drop-down lists
  • After you do that, click the “Set Matrices” button to get the desired matrix format
  • Now fetch the numbers in their fields
  • After you are done with the stuff, hit the calculate button


The best gauss jordan elimination calculator with steps does the following calculations:

  • Shows variables’ coefficients
  • Displays Gaussian elimination steps


What is the difference between Gauss elimination and Gauss Jordan method?

There exist only one difference between both. Where there the Gauss elimination generates the reduced echelon form of the given matrix, Gauss Jordan does produce only the echelon form. And our best gauss calculator will let you determine the reduced form which is more efficient than getting only the echelon format of the matrix.

Why do we use Gauss elimination method?

The row reduction strategy for solving linear equations systems is known as the Gaussian elimination method in mathematics. It’s made up of a series of operations on the associated coefficients matrix. This approach may also be used to estimate the following:

  • The supplied matrix’s rank.
  • A square matrix’s determinant
  • An invertible matrix’s inverse

Which is more efficient Gauss Jordan or Gauss Elimination?

Because it skips the process of backward substitution, the Gauss Jordan Method is more efficient than Gaussian Elimination techniques. And this is why our free Gauss jordan method calculator uses this particular technique to depict the outcomes straightaway.

What is Gaussian elimination used for in real life?

Robust Fingerprint Image Enhancement is an essential use of Gaussian elimination. The image is enhanced with a Gaussian filter. On mesh-connected processors, the SGE technique may also be used to solve linear equations. In scheduling algorithms, the Gaussian approach is also utilised.

When Gauss elimination method is used?

The most common use of Gauss elimination is to solve a set of linear algebraic problems.

What are advantages of Gauss elimination method?

  • This procedure is entirely trustworthy and fair.
  • It can solve many linear equations at the same time.

What is pitfalls of Gauss elimination method?

Close to zero determinant This indicates that there are no or an unlimited number of solutions. However, determining how near the determinant must be to zero in order to indicate ill-conditioning is problematic.

What is pivoting in Gaussian elimination?

Pivoting has the goal of turning an element above or below a leading one into a zero. The “pivot” or “pivot element” is a left-hand side of a matrix element for which the components above and below should be zero. Rounding mistakes are reduced by pivoting.

What is a pivot in matrix?

In the row-echelon form of a matrix, the pivot point is the location of a leading entry. A pivot column is one that has a pivot location in it.


Gauss jordan matrices have vast applications in various fields of education and technology. Due to such an importance, we have designed this best gaussian elimination calculator matrix so as to assist anyone analysing this particular technique to resolve equations.


From the source of wikipedia: Gaussian elimination, Row operations, Echelon form, Computing determinants, inverse of a matrix, Ranks 

From the source of khan academy: Matrix row operations, Solving linear systems with matrices

From the source of lumen learning: Gaussian Elimination, Augmented Matrix, System of Equations, Applications