Discuss the cases of Gaussian elimination method.

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Jason Olsen · Accepted Answer

First write down the given equations in the form of a matrix
Next by applying the row transformations, let’s reduce it in the form of a upper triangular matrix
i.e. the elements below the diagonal are zero
Next let’s form an equations from the upper triangular matrix.
By back substitution, we will find the values of x

Sean Becker · Accepted Answer

To perform row reduction on a matrix, one uses a sequence of elementary row operations to modify the matrix until the lower left-hand corner of the matrix is filled with zeros, as much as possible. There are three types of elementary row operations: Swapping two rows, Multiplying a row by a nonzero number, Adding a multiple of one row to another row. Using these operations, a matrix can always be transformed into an upper triangular matrix, and in fact one that is in row echelon form. Once all of the leading coefficients (the leftmost nonzero entry in each row) are 1, and every column containing a leading coefficient has zeros elsewhere, the matrix is said to be in reduced row echelon form. This final form is unique; in other words, it is independent of the sequence of row operations used. For example, in the following sequence of row operations (where two elementary operations on different rows are done at the first and third steps), the third and fourth matrices are the ones in row echelon form, and the final matrix is the unique reduced row echelon form. [131911−11311535]→[13190−2−2−80228]→[13190−2−2−80000]→[10−2−301140000] Using row operations to convert a matrix into reduced row echelon form is sometimes called Gauss–Jordan elimination. In this case, the term Gaussian elimination refers to the process until it has reached its upper triangular, or (unreduced) row echelon form. For computational reasons, when solving systems of linear equations, it is sometimes preferable to stop row operations before the matrix is completely reduced.

Discuss the cases of Gaussian elimination method.

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