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Gaussian Elimination: How It Can Be Used To Solve a System Of Linear Equations

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Gaussian elimination which is also known as row reduction, is an algorithm used in linear algebra to solve system of linear equation. These sequence of operations are understood by performing on the corresponding matrix of coefficients. This method is also used to find the rank of a matrix, for calculating the determinant of a matrix, and also to calculate the inverse of an invertible square matrix. Carl Friedrich Gauss is the person after whom this method is named, however these operations were known to Chinese mathematicians during 179 A.D. The performing of row reduction in a matrix is done where a sequence of elementary row operations are modified for the matrix until and unless the lower left hand corner of the matrix is filled with zeros, as much as possible. There are three types of elementary row operations, which are swapping of two rows, multiplication of a row by non-zero numbers and adding a multiple of one row to another row.

The elementary row operations help in transforming the matrix into an upper triangular matrix, which in fact is in row echelon form. Once all the leading coefficients ( leftmost non-zero entry in each row) are 1, and every column having a leading coefficient has zeros somewhere else, where the matrix is said to be in reduced row echelon form. The final form is unique, which means it is independent of the sequence of row operations performed. Use of row operations for conversion of matrix to reduced row echelon form is sometimes termed as Gauss-Jordan elimination. The term Gauss elimination is termed by some authors for referring the process till it has reached the upper triangular, or unreduced row echelon form. Due to computational reasons, while solving a system of linear equation, it is preferable to stop row operation before the complete reduction of the matrix.

There are three processes of elementary row operations, which can be performed on the rows of a matrix, which are swapping the position of two rows, multiplying a row by non-zero scalar, or addition of one row with the scalar multiplier of another. The association of matrix in a system of linear equation, do not change the solution set, thus if the goal is solving a system of linear equation, then use of these row operations help to make the problem easier.

The row echelon forms each row in a matrix, do not consist of only zeros, but the leading nonzero term is called the leading co-efficient of that specific row. Thus, if two leading coefficients are in the same column, then a row operation of type 3 is used to make one of the coefficients zero. Thus, by using the row swapping operation, it can always order the rows so that, every nonzero row, the leading coefficient is present in the right of the leading coefficient of the row above. This form is termed as the row echelon form, where the lower left part of the matrix contains only zeros, and all the zeros are present below the non-zero rows. The term echelon is used here because of the ranking of rows based on their size, with the largest number present at the top and the smallest is present at the bottom. Matrix is in reduced echelon form if the leading coefficients are equal to 1, which can be acquired by using the elementary row operation of type 2, and in every column, which contains a leading coefficient, all the other entries in that column are zero, which is achieved by performing elementary row operation of type 3.

System of equations is a group of linear equations with a lot of unknown factors that appear in various equations. To solve a system consisting for finding the value of unknown factors in a way that verifies all the equations that make the system. Presence of a single solution, one value for each unknown factor is termed as consistent independent system, if there are various systems ( systems having infinite number of solutions), the system is said to be consistent dependent system. However, if there are no solutions, which happens if there are two or more equations unverifiable at the same time, then that system is termed as inconsistent system.

For example, y = 0

        2x + y = 0

         2x + y = 2

This system is inconsistent, because obtaining the solution x = 0 from the second equation and from the third, x = 1. This section is solved by Gaussian Elimination method, that has simple elemental operations, both in rows and columns of the augmented matrix for obtaining echelon form or the reduced echelon form.

Suppose, the goal is to find the set of solutions for the following system of linear equation

2x + y – z = 8 (L1)

-3x – y + 2z = -11 (L2)

-2x + y + 2z = -3 (L3)

The process to deal with systems in terms of equation, does not deal with systems but instead uses augmented matrix, is much more useful for computer manipulation. The row reduction process can be summarized like, elimination of all x terms below L1, then elimination of all y terms from the equation below L2, which will make the system triangular form. Back substitution method is then used for the unknown components within the system. After elimination of y from the third row, the result is a system of linear equation in triangular form, and the first part of the algorithm is complete. Computational point of view provides faster ways in solving the variable in reverse order, which is known as back substitution. The solution after performance of row reduction is z = -1 , y = 3 and x = 2, which is an unique solution to the original system of equations.

Stopping the matrix in echelon form, it can be continued until the matrix is in reduced row echelon form. The process of row reduction until the matrix is reduced is sometimes referred to s Gauss-Jordan elimination, for distinguishing it to stop after the system reaches the echelon form.

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