1.6. ELEMENTARY MATRICES 19

As an example of why these elementary matrices are interesting, consider the following.Letting ri be the row vector of all zeros except for a 1 in the ith slot, r2

r1r3

 a b c dx y z wf g h i

=

 x y z wa b c df g h i

 .

A 3×4 matrix was multiplied on the left by an elementary matrix which was obtained fromrow operation 1 applied to switching the first two rows of the identity matrix. This resultedin applying the operation 1 to the given matrix. This is what happens in general.

Now consider what these elementary matrices look like. They are obtained from switch-ing a couple of rows of the identity matrix. First Pi j, which involves switching row i androw j of the identity where Let i < j. Then, as above, Pi j =

r1...r j...ri...rn

where

r j = (0 · · ·1 · · ·0)

with the 1 in the jth position from the left.For Pi j this matrix which involves switching the i and j rows of the identity. Now

consider what this does to a column vector.

r1...r j...ri...rn





v1...vi...

v j...

vn

=



v1...

v j...vi...

vn

.

Now we try multiplication of a matrix on the left by this elementary matrix Pi j. Thus,

Pi j



a11 a12 · · · · · · · · · · · · a1p...

......

ai1 ai2 · · · · · · · · · · · · aip...

......

a j1 a j2 · · · · · · · · · · · · a jp...

......

an1 an2 · · · · · · · · · · · · anp

.