Page 65 - A Course in Linear Algebra with Applications

2.3: Elementary Matrices 49

Hence E^E 2E\A = B where

*-(!i).*-(Y:)'*-(i _ 1 / ?)

Column operations
Just as for rows, there are three types of elementary col-
umn operation, namely:
(a) interchange columns i and j , ( C{ «-> Cj);
(b) add c times column j to column i where c is a scalar,
(Ci + cCj);
(c) multiply column i by a non-zero scalar c, ( cCi).
(The reader is warned, however, that column operations can-
not in general be applied to the augmented matrix of a linear
system without changing the solutions of the system.)
The effect of applying an elementary column operation
to a matrix is simulated by right multiplication by a suitable
elementary matrix. But there is one important difference from
the row case. In order to perform the operation Ci + cCj to a
matrix A one multiplies on the right by the elementary matrix
whose (j, i) element is c. For example, let

12
E=( l *) and A=( an ° V
\c 1J \a 2i a 22J

Then
_ / i n
AE +ca i 2 a 12
\a 2i + ca 22 a 22

Thus E performs the column operation C\ + 2C 2 and not
+ 2C\. By multiplying a matrix on the right by suitable
C 2
sequences of elementary matrices, a matrix can be put in col-
umn echelon form or in reduced column echelon form; these