A row operation on a rectangular matrix with elements in a ring is an operation of replacing it with another matrix of the same dimensions which is of one of the following types

interchanging of two rows of the matrix

rescaling one row by multiplying it with a nonzero element

adding a multiple of one row to some other row.

Similarly, a column operation is either interchanging two columns, rescaling one column by multiplying it with a nonzero element or adding a multiple of one column to some other column.

Properties

Each row operation can be achieved by multiplying the matrix from the left by an elementary matrix.

If the ground ring is a field then each row operation is invertible.