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In mathematics, the Gaussian elimination method is known as the row reduction algorithm for solving linear equations systems. It consists of a sequence of operations performed on the corresponding matrix of coefficients. We can also use this method to estimate either of the following: The rank of the given matrix.
Highlights. The Gauss-Jordan method is similar to the Gaussian elimination process, except that the entries both above and below each pivot are zeroed out. After performing Gaussian elimination on a matrix, the result is in row echelon form, while the result after the Gauss-Jordan method is in reduced row echelon form.
There are three steps involved in Gaussian elimination: Convert the system of equations to an augmented matrix. Put the matrix in upper triangular form. Solve for the variables starting with the last row and working your way up.
Permitted actions There are only two actions you can do in standard Gaussian elimination: they are: swap two rows; add (or subtract) a multiple of one row to a row below it.
To perform Gauss-Jordan Elimination: Swap the rows so that all rows with all zero entries are on the bottom. Swap the rows so that the row with the largest, leftmost nonzero entry is on top. Multiply the top row by a scalar so that top rows leading entry becomes 1.
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How is a set of equations solved numerically by Gaussian elimination method? Forward Elimination of Unknowns: In this step, the unknown is eliminated in each equation starting with the first equation. This way, the equations are reduced to one equation and one unknown in each equation.
The difference between Gauss elimination and the Gauss Jordan method is that one produces a matrix in row echelon form while the other produces a matrix in reduced-row echelon form.
As for the time efficiency, in the article of System of Linear Equations, Gaussian Elimination, Gharib et al. [1] showed that the Gaussian elimination is more efficient over the Gauss-Jordan reduction in solving systems of linear equations.

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