HOMEWORK 4  SOLUTIONS
1. Easy, just apply the techniques from class, namely cofactor expansion along any row,
cofactor expansion down any column, and row reduction. The answer is

48

34
i
.
2. (GRADED  MEDIUM)
(a) tr(
CAC

1
) = tr(
C

1
CA
) = tr(
A
).
(b) A simple computation:
det(
CAC

1
) = det(
C
1
)det(
A
)det(
C
)
= det(
A
)det(
C
)det(
C

1
)
= det(
A
)det(
CC

1
)
= det(
A
)det(
I
)
= det(
A
)
(c) First observe:
Ax
= 0
→
AC

1
(
Cx
) = 0
→
CAC

1
(
Cx
) = 0
→
AC

1
(
Cx
) =
C

1
0
→
Ax
= 0
Thus
x
∈
ker(
L
A
) iﬀ
Cx
∈
ker(
L
CAC

1
). Since
C
is invertible,
L
C
is an isomor
phism, and so it’s not hard to see that
L
C
±
ker(
L
A
)
: ker(
L
A
)
→
ker(
L
CAC

1
) is an
isomorphism. Thus unravelling the deﬁnitions gives us:
null(
A
) = null(
L
A
)
= dim(ker(
L
A
))
= dim(ker(
L
CAC

1
))
= null(
L
CAC

1
)
= null(
CAC

1
)
(d) This follows from part (c):
rank(
A
) = rank(
L
A
)
=
n

null(
L
A
)
=
n

null(
L
CAC

1
)
= rank(
L
CAC

1
)
= rank(
CAC

1
)
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document3. (a) No. Let
T
= id
R
and
U
=

id
R
. Then clearly
T
and
U
are isomorphisms
R
→
R
but
T
+
U
is the zero map, which is clearly not an isomorphism.
(b) Yes. (
αT
)(
x
) = 0
→
α
(
T
(
x
)) = 0
→
T
(
x
) = 0
→
x
= 0 where the ﬁrst impli
cation follows by deﬁnition of the scalar multiple of a transformation, the second
by the fact that
α
6
= 0, and the last by the fact that
T
is an isomorphism. Since
T
is an isomorphism, it’s a map between vector spaces of the same dimension,
hence
αT
is a map between spaces of the same dimension. Since the only vector
it maps to zero is zero,
αT
is an isomorphism.
4. (a) It’s easy to see that
UT
has an inverse, in fact the formula for it is given to us:
(
UT
)(
T

1
U

1
) =
UTT

1
U

1
=
U
◦
id
W
◦
U

1
=
UU

1
= id
X
(
T

1
U

1
)(
UT
) =
T

1
U

1
UT
=
T

1
◦
id
W
◦
T
=
T

1
T
= id
V
(b) Do a similar computation to the one above.
(c)
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 GUREVITCH
 Math, Linear Algebra, Vector Space, u2, Diagonal matrix, αt

Click to edit the document details