The term Frobenius matrix may also be used for an alternative matrix form that differs from an Identity matrix only in the elements of a single row preceding the diagonal entry of that row (as opposed to the above definition which has the matrix differing from the identity matrix in a single column below the diagonal).
An alternative name for this latter form of Frobenius matrices is Gauss transformation matrix, after Carl Friedrich Gauss.
If a matrix is multiplied from the left (left multiplied) with a Gauss transformation matrix, a linear combination of the preceding rows is added to the given row of the matrix (in the example shown above, a linear combination of rows 1 and 2 will be added to row 3).
Multiplication with the inverse matrix subtracts the corresponding linear combination from the given row.
This corresponds to one of the elementary operations of Gaussian elimination (besides the operation of transposing the rows and multiplying a row with a scalar multiple).