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2.3: Elementary Matrices 47

2.3 Elementary Matrices

An nxn matrix is called elementary if it is obtained from
the identity matrix I n in one of three ways:

(a) interchange rows i and j where i ^ ;
j
j
(b) insert a scalar c as the (i,j) entry where % ^ ;
(c) put a non-zero scalar c in the (i, i) position.
Example 2.3.1
Write down all the possible types of elementary 2x2 matrices.
These are the elementary matrices that arise from the matrix

12 = t h e y are
( o i ) '

l
Ei=[ I ),E =( l 0 [ ) , * * = ( I I
2

and
* - « s : • * - ( * ° e
)

Here c is a scalar which must be non-zero in the case of E4
and E 5.

The significance of elementary matrices from our point of
view lies in the fact that when we premultiply a matrix by an
elementary matrix, the effect is to perform an elementary row
operation on the matrix. For example, with the matrix

A _ 1 i n ^12
«21 «22

and elementary matrices listed in Example 2.3.1, we have

c
EA=( a21 a22 \ EA=( ail + Ca21 a i2 + «22 N
U i i a i 2 / ' 2 V a 2i a 22