Elementary row operation

An elementary row operation interchanges two rows of a matrix, multiplies one row by a nonzero scalar, or adds a scalar multiple of another row to it. These invertible operations are used in Gaussian elimination. For a square matrix, interchange changes the sign of its determinant, multiplication scales its determinant by the same scalar, and adding another row leaves its determinant unchanged, by multilinearity and alternation.

An elementary matrix is obtained by applying one elementary row operation to the identity matrix. Multiplying $A$ by it on the left performs the same row operation on $A$. Such a matrix is invertible: reverse a row swap, reverse a nonzero row scaling by its reciprocal, or reverse adding a multiple of another row by adding its negative. Right multiplication gives an elementary column operation. In particular these operations preserve matrix rank.