1.6. ELEMENTARY MATRICES 25
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, 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
.