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midterm1solutions_Math 113

# midterm1solutions_Math 113 - Math 113 Section 5 Fall 2010...

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Math 113 Section 5 Midterm #1 Solutions Fall 2010 Wednesday, September 22 1. Determine which of the following sets with 2-to-1 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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midterm1solutions_Math 113 - Math 113 Section 5 Fall 2010...

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