Math 113 Section 5
Midterm #1 Solutions
Fall 2010
Wednesday, September 22
1.
Determine which of the following sets with 2to1 operations form groups. For those
that are groups, prove that they are groups. For those that are not groups, describe
why they are not groups.
(a)
(
Z
,
·
) is not a group because not every integer has a multiplicative inverse which
is also an integer. For example,
1
2
/
∈
Z
.
(b)
(
Q
>
0
,
·
) is a group.
Closure: If
a
b
>
0 and
a
0
b
0
>
0 then
aa
0
bb
0
>
0.
Associativity follows from associativity of multiplication in
R
.
Identity: 1
∈
Q
>
0
.
Inverses: If
a
b
>
0 then
b
a
>
0 and
a
b
b
a
= 1.
(c)
(
{
M
∈
GL
(
n,
R
)

M
is diagonal
}
,
·
) is a group.
Closure:
Let
A
and
B
be diagonal
n
×
n
matrices with entries
a
1
, a
2
, . . . , a
n
and
b
1
, b
2
, . . . , b
n
respectively. Then
AB
is diagonal with entries
a
1
b
1
, a
2
b
2
, . . . a
n
b
n
.
Associativity follows from associativity of matrix multiplication. Note that for
n
×
n
matrices, (
AB
)
C
and
A
(
BC
) are always defined.
Identity: The identity matrix is diagonal.
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 Summer '11
 gelfand
 Chemistry, Diagonal matrix, Inverses

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