Download (6) The determinant of the 2 × 2 matrix A = abc dis detA = ad

Survey
yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project

Document related concepts
no text concepts found
Transcript 
2. GROUPS

a b
(6) The determinant of the 2 ⇥ 2 matrix A =
is det A = ad
c d
Consider
⇢
a b
GL(2, R) =
a, b, c, d 2 R, ad bc 6= 0
c d
with multiplication



a1 b1 a2 b2
a a + b1c2 a1b2 + b1d2
= 1 2
.
c1 d1 c2 d2
c1a2 + d1c2 c1b2 + d1d2
29
bc.
Multiplication is closed in GL(2, R) since
 det(AB) = (det A)(det
 B). Associa1 0
a b
tivity is true, but messy; the identity is
; the inverse of
is
0 1
c d
 d

b
1
d
b
ad bc ad bc =
.
c
a
c
a
ad
bc
ad bc ad bc
This is the general linear group of 2 ⇥ 2 matricies over R. Since






1 2 2 3
10 13
2 3 1 2
11 16
=
and
=
,
3 4 4 5
22 29
4 5 3 4
19 28
GL(2, R) is non-Abelian.
The set of all 2 ⇥ 2 matrices over R with matrix multiplication is not a group
since matrices with 0 determinant do not have inverses.
(7) Consider Zn with multiplication modulo n. Are there multiplicative inverses? If so, we have a group.
Suppose a 2 Zn and ax mod n = 1 has a solution (i.e., a has an inverse). Then
ax = qn + 1 for some q 2 Z =) ax + n( q) = 1 =) a and n are relatively
prime by Theorem 0.2.
Now suppose a is relatively prime to n. Then, again by Theorem 0.2,
9 s, t 2 Z 3 as+nt = 1 =) as = ( t)n+1 =) as mod n = 1 =) s = a 1.
Thus we have proven Page 24 # 11:
Related documents