Zeilentausch

Zeilentausch, in English often called row interchange or row swap, is an elementary row operation in linear algebra that interchanges two rows of a matrix. Formally, for an m-by-n matrix A and distinct row indices i and j, the result A' is identical to A except that row i of A' equals row j of A and row j of A' equals row i of A.

In matrix notation, there exists an elementary matrix Eij obtained from the identity matrix by swapping its i-th and j-th rows; left-multiplying A by Eij yields A' = Eij A. This operation preserves the overall row space and the rank of the matrix. If A is square, swapping two rows multiplies the determinant by -1, so det(A') = -det(A) (a sign change reflecting the swap).

Zeilentausch is a standard tool in Gaussian elimination and other row-reduction procedures. It is used to move a nonzero element into a pivot position or to avoid division by zero, thereby enabling subsequent elimination steps.

Example: Consider A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]. Swapping rows 1 and 3 yields A' = [[7, 8, 9], [4, 5, 6], [1, 2, 3]].

See also: elementary row operations; Gaussian elimination; determinant properties. Zeilentausch is one of three fundamental elementary row operations, along with adding a multiple of one row to another and multiplying a row by a nonzero scalar.