# Gauss Jordan Elimination Method Step By Step Pdf Writer ## Gaussian elimination

In reduced row echelon form, each successive row of the matrix has less dependencies than the previous, so solving systems of equations is a much easier task. The idea behind row reduction is to convert the matrix into an "equivalent" version in order to simplify certain matrix computations. Its two main purposes are to solve system of linear equations and calculate the inverse of a matrix. Carl Friedrich Gauss championed the use of row reduction, to the extent that it is commonly called Gaussian elimination.

It was further popularized by Wilhelm Jordan, who attached his name to the process by which row reduction is used to compute matrix inverses, Gauss-Jordan elimination. A system of equations can be represented in a couple of different matrix forms.

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One way is to realize the system as the matrix multiplication of the coefficients in the system and the column vector of its variables. The square matrix is called the coefficient matrix because it consists of the coefficients of the variables in the system of equations:. An alternate representation called an augmented matrix is created by stitching the columns of matrices together and divided by a vertical bar.

## Explanation

The coefficient matrix is placed on the left of this vertical bar, while the constants on the right-hand side of each equation are placed on the right of the vertical bar:. The matrices that represent these systems can be manipulated in such a way as to provide easy-to-read solutions. This manipulation is called row reduction. Row reduction techniques transform the matrix into reduced row echelon form without changing the solutions to the system.

To convert any matrix to its reduced row echelon form, Gauss-Jordan elimination is performed. There are three elementary row operations used to achieve reduced row echelon form:.

The top left element is a pivot, so the rest of the elements in the first column must be 0. This can be done by subtracting the first row from the second row.

Furthermore, the first row can be added to the third row to obtain the necessary 0s in the first column:. The top and bottom elements in the second column can be made 0 with the appropriate row operations:.

## Gauss-Jordan Elimination

If a column has a pivot, then the rest of the elements in the column will be 0 0 0. Rows that consist of only zeroes are in the bottom of the matrix. There are three elementary row operations used to achieve reduced row echelon form: Switch two rows. Multiply a row by any non-zero constant. Add a scalar multiple of one row to any other row.

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