Understanding the Reversibility of Elementary Row Operations in Linear Algebra

Every elementary row operation is reversible

In linear algebra, elementary row operations refer to a set of three operations that can be performed on the rows of a matrix without changing its row space

In linear algebra, elementary row operations refer to a set of three operations that can be performed on the rows of a matrix without changing its row space. These operations are:

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

The statement “Every elementary row operation is reversible” means that for any given matrix, if we perform an elementary row operation on it, we can always reverse that operation using another elementary row operation to obtain the original matrix.

Let’s examine each of the elementary row operations and see why they are reversible:

1. Swapping two rows: Swapping two rows of a matrix can be undone simply by swapping those same rows again. For example, if we swap the first and second rows of a matrix, we can revert back to the original matrix by performing the same swap operation.

2. Multiplying a row by a non-zero scalar: Multiplying a row by a non-zero scalar and then dividing the result by the same non-zero scalar will give us the original row. This is because multiplying and dividing by the same non-zero scalar cancels out the effect of the multiplication. For example, if we multiply the second row of a matrix by 2, we can reverse the operation by dividing the resulting row by 2.

3. Adding a multiple of one row to another row: Adding a multiple of one row to another row can be undone by subtracting the same multiple of the same row from the other row. This is because adding and subtracting yield opposite effects. For example, if we add twice the first row to the second row, we can reverse the operation by subtracting twice the first row from the second row.

In conclusion, every elementary row operation is reversible because we can always perform another elementary row operation (or a combination of them) to restore the original matrix. These reversible operations are fundamental in solving systems of linear equations and performing row reduction to find solutions or to transform matrices into a desired form.

More Answers: