Reduced row echelon form of matrix

[bimal]

How do you know, when you have to stop row-equivalent operations when you are trying to get a 'reduced row-echelon' form of a given matrix. Is it necessary to have all the columns with pivot element as 1 and rest as 0? do you need to continue the operation if you already have a all 0 row? I want to use the number of columns with pivot element as 1 to determine the linearly indendent columns.

 

[Aneleh]

A reduced row echelon form matrix has "1" as the leading entry in each nonzero row, and each leading 1 is the only nonzero entry in its column. As well each leading entry of a row is in a column to the right of the leading entry of the row above it.

 

[tacman]

The reduced row echelon form of a matrix is a unique representation of the matrix that is obtained by performing row operations until the matrix satisfies certain conditions. These conditions include having the leading coefficient (pivot element) of each row to be 1, and all elements below the leading coefficient to be 0. In addition, the leading coefficient of each row must be to the right of the leading coefficient of the row above it.

To know when to stop row-equivalent operations, we can look at the leading coefficients of each row. Once all the leading coefficients are 1 and are in the correct positions, we can stop the operations and the matrix will be in reduced row echelon form.

It is necessary to have all the columns with a pivot element as 1 and all other elements in that column as 0 in order for the matrix to be in reduced row echelon form. This is because the leading coefficient of each row represents the pivot element for that row, and having all other elements in that column as 0 ensures that the matrix is in its simplest form.

If we encounter a row with all 0s, this indicates that the row is redundant and does not add any new information to the matrix. In this case, we can omit this row and continue the operations on the remaining rows until we reach the desired reduced row echelon form.

Using the number of columns with a pivot element as 1 is a valid way to determine the linearly independent columns of a matrix. This is because each pivot element represents a linearly independent variable in the system of equations represented by the matrix. Therefore, the number of columns with pivot elements will equal the number of linearly independent variables in the system.

In summary, the reduced row echelon form of a matrix is obtained by performing row operations until the matrix satisfies certain conditions, including having all leading coefficients as 1 and all other elements in that column as 0. It is not necessary to continue operations if we encounter a row with all 0s, as this row is redundant. The number of columns with pivot elements can be used to determine the linearly independent columns of the matrix.