Every elementary row operation is reversible.
Every elementary row operation in linear algebra is reversible
Every elementary row operation in linear algebra is reversible.
There are three types of elementary row operations:
1. Swapping Rows: This operation involves swapping the positions of two rows in a matrix. For example, if we have a matrix [A], and we swap rows i and j, we obtain a new matrix [B]. We can easily reverse this operation by swapping the same two rows again, i.e., swapping rows i and j in [B] to obtain [A] again.
2. Multiplying a Row by a Nonzero Scalar: This operation involves multiplying all the elements of a row in a matrix by a nonzero scalar. For example, if we have a matrix [A], and we multiply row i by a nonzero scalar k, we obtain a new matrix [B]. To reverse this operation, we divide all the elements of the same row in [B] by the same nonzero scalar k, resulting in [A] again.
3. Adding a Multiple of One Row to Another Row: This operation involves adding a multiple of one row to another row in a matrix. For example, if we have a matrix [A], and we add k times row i to row j, we obtain a new matrix [B]. To reverse this operation, we subtract the same multiple (k times row i) from row j in [B], resulting in [A] again.
In summary, each elementary row operation has an operation that can be performed to reverse its effect, allowing us to easily undo any changes made to a matrix. This reversibility is useful in many applications, such as solving systems of linear equations, finding inverses of matrices, and performing row reduction to find echelon form or reduced echelon form.
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