1.6. ELEMENTARY MATRICES 29
=
a11 a12 · · · a1p...
......
ai1 ai2 · · · aip...
......
a j2 + cai1 a j2 + cai2 · · · a jp + caip...
......
an1 an2 · · · anp
.
The case where i > j is similar. This proves the following lemma in which, as above, theith row of the identity is ri.
Lemma 1.6.5 Let E (c× i+ j) denote the elementary matrix obtained from I by replac-ing the jth row of the identity r j with cri +r j. Letting the kth row of A be ak,
E (c× i+ j)A = B
where B has the same rows as A except the jth row of B is cai +a j.
The above lemmas are summarized in the following theorem.
Theorem 1.6.6 To perform any of the three row operations on a matrix A it sufficesto do the row operation on the identity matrix, obtaining an elementary matrix E, and thentake the product, EA. In addition to this, the following identities hold for the elementarymatrices described above.
E (c× i+ j)E (−c× i+ j) = E (−c× i+ j)E (c× i+ j) = I. (1.8)
E (c, i)E(c−1, i
)= E
(c−1, i
)E (c, i) = I. (1.9)
Pi jPi j = I. (1.10)
Proof: Consider (1.8). Starting with I and taking −c times the ith row added to the jth
yields E (−c× i+ j) which differs from I only in the jth row. Now multiplying on the leftby E (c× i+ j) takes c times the ith row and adds to the jth thus restoring the jth row to itsoriginal state. Thus E (c× i+ j)E (−c× i+ j) = I. Similarly E (−c× i+ j)E (c× i+ j) =I. The reasoning is similar for (1.9) and (1.10). ■
Each of these elementary matrices has a significant geometric significance. The effectof doing E
( 12 ×3+1
)shears the box in one direction. Of course there would be corre-
sponding shears in the other directions also. Note that this does not change the volume.You should think about the geometric effect of the other elementary matrices on a box.
Definition 1.6.7 For an n× n matrix A, an n× n matrix B which has the propertythat AB = BA = I is denoted by A−1. Such a matrix is called an inverse. When A has aninverse, it is called invertible.
The following lemma says that if a matrix acts like an inverse, then it is the inverse.Also, the product of invertible matrices is invertible.