Observe that multiplication of the matrix Aby an elementary matrix E produces
the following elementary transformations of the matrix A.
E 1 A=
d e f
a b c
g h i
where the first two rows of Aare interchanged,
E 2 A=
d e f
g h i
a b c
where simultaneously row 2 is moved to row 1, row 3 is moved to row 2 and row 1
is moved to row 3,
E 3 A=
3 a 3 b 3 c
d e f
g h i
where row 1 is multiplied by the scalar 3, and
E 4 A=
a+ 3d b + 3e c + 3 f
d e f
g h i
where row 2 of Ais multiplied by 3 and the result is added to row 1. Observe that
the elementary matrices E 1 , E 2 , E 3 and E 4 are obtained by performing elementary row
operations on the identity matrix. When one of these elementary matrices multiplies
the matrix Ait has the same effect as performing the corresponding elementary row
operation on the matrix A.
Let the product of successive elementary row transformations be denoted by
EkEk− 1 ···E 3 E 2 E 1 =P.
Similarly, one can define the product of successive elementary column transforma-
tions by
E 1 E 2 E 3 ···Em=Q.
The equivalence of two matrices Aand B is defined as follows. Let P and Qdenote,
respectively, the product of successive elementary row and column transformations
as defined above. If B=P A, then Bis said to be row equivalent to A. If B=AQ, then
