Row Reduction Algorithm | Solving Linear Equations Using Augmented Matrices

The row reduction algorithm applies only to augmented matrices for a linear system.

The row reduction algorithm, also known as Gaussian elimination, is a method used to solve systems of linear equations

The row reduction algorithm, also known as Gaussian elimination, is a method used to solve systems of linear equations. It is not limited to only augmented matrices, but can be applied to any matrix representation of a system of linear equations.

However, the use of augmented matrices is a common and convenient way to represent linear systems because it allows us to combine the coefficient matrix and the constants vector into a single matrix. This augmented matrix makes it easier to perform the row reduction algorithm.

In the row reduction algorithm, the goal is to transform the augmented matrix into its row-echelon form or reduced row-echelon form. Row-echelon form is a matrix where all rows containing non-zero elements have their left-most non-zero element equal to 1, and each pivot element (the leading non-zero element in a row) is to the right of the pivot element in the previous row.

Reduced row-echelon form is a further simplification of the row-echelon form, where all entries above and below the pivot elements are zero. In the reduced row-echelon form, each pivot element is the only non-zero entry in its column.

By applying row operations, such as scaling a row, swapping rows, or adding a multiple of one row to another, the augmented matrix can be transformed into row-echelon or reduced row-echelon form. The resulting matrix can then be used to determine the solution(s) to the linear system.

In conclusion, while the row reduction algorithm is not limited to augmented matrices, they are commonly used because they provide a compact and convenient representation for solving linear systems.

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