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The Nuts and Bolts of Proofs, 4th Edition
book

The Nuts and Bolts of Proofs, 4th Edition

by Antonella Cupillari
January 2011
Beginner content levelBeginner
296 pages
11h 43m
English
Academic Press
Content preview from The Nuts and Bolts of Proofs, 4th Edition
(Continued )
Let A and B be any two elements of M
2
, define their sum as
A + B =
a
11
a
12
a
21
a
22

+
b
11
b
12
b
21
b
22

=
a
11
+ b
11
a
12
+ b
12
a
21
+ b
21
a
22
+ b
22

:
Prove that this operation is associative.
Proof: Let A, B, and C be any elements of M
2
.Is(A + B)+C = A +(B + C)?
Let A =
a
11
a
22
a
21
a
12

, B =
b
11
b
12
b
21
b
22

, and C =
c
11
c
12
c
21
c
22

: Then
ðA + BÞ+ C =
a
11
+ b
11
a
12
+ b
12
a
21
+ b
21
a
22
+ b
22

+
c
11
c
12
c
21
c
22

=
ða
11
+ b
11
Þ+ c
11
ða
12
+ b
12
Þ+ c
12
ða
21
+ b
21
Þ+ c
21
ða
22
+ b
22
Þ+ c
22

:
and
A + ðB + CÞ=
a
11
a
12
a
21
a
22

+
b
11
+ c
11
b
12
+ c
12
b
21
+ c
21
b
22
+ c
22

=
a
11
+ ðb
11
+ c
11
Þ a
12
+ ðb
12
+ c
12
Þ
a
21
+ ðb
21
+ c
21
Þ a
22
+ ðb
22
+ c
22
Þ

:
All the numbers used to build the matrices are real numbers, therefore, ...
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Publisher Resources

ISBN: 9780123822178