Solutions to Homework 7.
Math 110, Fall 2006.
Prob 3.1.3.
To ﬁnd the inverses of the elementary matrices, we must simply undo the corresponding
elementary operations:
(
a
)
0
0
1
0
1
0
1
0
0
,
(
b
)
1
0
0
0
1
/
3
0
0
0
1
,
(
c
)
1
0
0
0
1
0
2
0
1
.
Prob 3.1.5.
If
E
is an elementary matrix of type I corresponding to the swap or rows
i
and
J
, then
E
has 1 in positions (
i, j
) and (
j, i
), zeros in positions (
i, i
) and (
j, j
), and the other entries exactly as in the
identity matrix of the appropriate size. But then
E
is symmetric, i.e., its transpose is
E
itself. Likewise, if
E
is of type II, then its transpose is itself. If
E
is of type III and corresponds to adding row
j
multiplied by
c
to row
i
, then it looks like the identity matrix except for the entry
c
in position (
i, j
). Then its transpose
has
c
in position (
j, i
), which corresponds to adding row
i
multiplied by
c
to row
j
.
So, in all three cases
E
is an elementary matrix if and only if so is
E
t
.
Prob 3.2.2.
(a) 2, (b) 3, (c) 2, (d) 1, (e) 3, (f) 3, (g) 1.
Prob 3.2.6.
(a)
T
is invertible. Writing the matrix of
T
in the standard basis for
P
2
(IR) and inverting it, we get
T

1
(
ax
2
+
bx
+
c
) =

ax
2

(4
a
+
b
)
x

(10
a
+ 2
b
+
c
)
.
(b)
T
is not invertible, since it has a nontrivial kernel, for example,
T
1 = 0.