Math 3124
Tuesday, August 30
August 30 Ungraded Homework
Problem 5.16 on page 34
Let
G
denote the set of all 2
×
2 real matrices
A
with 0
=
det
(
A
)
∈
Q
. Prove that
G
is a group with respect to matrix multiplication. (You may assume
that matrix multiplication is associative.
But check closure (so matrix multiplication is
an operation on
G
), and the existence of an identity element and inverse elements very
carefully.) Is
G
abelian?
Let
I
denote the identity 2
×
2 matrix (1’s on the main diagonal and 0’s elsewhere). We will
want to use the wellknown property that det
(
AB
) =
det
(
A
)
det
(
B
)
for any 2
×
2 matrices
A
,
B
. If
A
,
B
∈
G
, then
AB
is also matrix with real entries. Since det
(
AB
) =
det
(
A
)
det
(
B
)
and
the product of two nonzero rational numbers is a nonzero rational number, we see that 0
=
det
(
AB
)
∈
Q
and we deduce that
AB
∈
G
. Thus we have closure and matrix multiplication
is indeed an operation on
G
.
Next matrix multiplication is associative (we are allowed
to assume this). Furthermore
I
∈
G
and is the identity for
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 Fall '08
 PARRY
 Algebra, Multiplication, Matrices, Ring

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