A square matrix obtained from the identity matrix I by an elementary row operation. Thus there are three types of elementary matrix. Examples of each type are: The matrix (i) is obtained from I by interchanging the second and fifth rows, matrix (ii) by multiplying the third row by −3, and matrix (iii) by adding 4 times the fifth row to the second row. Pre-multiplication of an m×n matrix A by an m×m elementary matrix produces the result of the corresponding row operation on A.
Alternatively, an elementary matrix can be seen as one obtained from the identity matrix by an elementary column operation; and post-multiplication of an m×n matrix A by an n×n elementary matrix produces the result of the corresponding column operation on A.