Elementary Row Operations And Row Echelon Form: Properties And Optimization

If an elementary row op is applied to a matrix that is in row echelon form, the resulting matrix will still be in row echelon form.

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Yes, this statement is true. An elementary row operation is a manipulation of the rows of a matrix, and there are three types of these operations:

1. Swapping any two rows.
2. Multiplying any row by a non-zero scalar.
3. Adding a multiple of one row to another row.

A matrix is said to be in row echelon form if:

1. All rows that contain only zeros are at the bottom of the matrix.
2. For each row that contains at least one non-zero entry, the first non-zero entry is called a pivot. The position of the pivot in each row is always to the right of the position of the pivot in the row above it.

Now, suppose that we apply an elementary row operation to a matrix that is already in row echelon form. Let us consider each type of operation:

1. Swapping any two rows: This operation does not change the fact that all rows that contain only zeros are at the bottom of the matrix, and does not change the position of the pivots.
2. Multiplying any row by a non-zero scalar: This operation does not change the fact that all rows that contain only zeros are at the bottom of the matrix, and does not change the position of the pivots.
3. Adding a multiple of one row to another row: This operation does not change the fact that all rows that contain only zeros are at the bottom of the matrix. Also, it only changes the entries of the rows below the row that is being used for the operation, and therefore does not affect the position of the pivots.

Therefore, we can say that if an elementary row op is applied to a matrix that is in row echelon form, the resulting matrix will still be in row echelon form.

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