2
PROFESSOR STEVEN MILLER: BROWN UNIVERSITY: SPRING 2007
Page 27, #10: True or false? Give a specific counterexample when false. (a)
If columns 1 and 3 of
B
are the same, so are columns 1 and 3 of
AB
. (b) If
rows 1 and 3 of
B
are the same, so are rows 1 and 3 of
AB
. (c) If rows 1
and 3 of
A
are the same, so are rows 1 and 3 of
AB
. (d)
(
AB
)
2
=
A
2
B
2
.
Page 27, #12: The product of two lower triangular matrices is lower trian
gular. Confirm this with a
3
×
3
example, and then explain how it follows
from the laws of matrix multiplication.
Page 28, # 20: The matrix that rotates the
x

y
plane by an angle
θ
is
A
(
θ
) =
cos
θ

sin
θ
sin
θ
cos
θ
¶
. Verify that
A
(
θ
1
)
A
(
θ
2
) =
A
(
θ
1
+
θ
2
)
from the
identities for
cos(
θ
1
+
θ
2
)
and
sin(
θ
1
+
θ
2
)
. What is
A
(
θ
)
times
A
(

θ
)
?
Page 40, #5: Factor
A
into
LU
and write down the upper triangular system
Ux
=
c
which appears after elimination, where
Ax
=
2
3
3
0
5
7
6
9
8
u
v
w
=
2
2
5
.
Page 42, # 25: When zero appears in a pivot position,
A
=
LU
is not
possible. Show why these are both impossible:
0
1
2
3
¶
=
1
0
‘
1
¶
d
e
0
f
¶
,
1
1
0
1
1
2
1
2
1
=
1
0
0
‘
1
0
m
n
1
d
e
g
0
f
h
0
0
i
.
Page 43, # 29: Compute
L
and
U
for the symmetric matrix
A
=
a
a
a
a
a
b
b
b
a
b
c
c
a
b
c
d
.
Page 44, # 41: How many exchanges will permute
(5
,
4
,
3
,
2
,
1)
back to
(1
,
2
,
3
,
4
,
5)
? How may exchanges to change
(6
,
5
,
4
,
3
,
2
,
1)
to
(1
,
2
,
3
,
4
,
5
,
6)
?
One is even and one is odd. For
(
n, . . . ,
1)
to
(1
, . . . , n
)
, show that
n
= 100