hw21 - Show that the usual formula yields the inverse of A...

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Modern Algebra 1 Homework 10.1 Spring 2010 Due April 7 Exercise 1. Let G be a group and let H,K C G . Prove that if K H and G/K is cyclic, then G/H is cyclic. [ Suggestion: Use the Third Isomorphism Theorem and the fact (proven in earlier homework) that a quotient of a cyclic group is cyclic.] Exercise 2. Let R be a commutative ring (with unity) and let M 2 ( R ) = ±² a b c d ³´ ´ ´ ´ a,b,c,d R µ . a. Show that if A = ² a b c d ³ M 2 ( R ) × then det( A ) = ad - bc R × . [ Hint: The result of exercise 3.2.6.a is valid for any commutative ring.] b. Show that if A = ² a b c d ³ M 2 ( R ) and det( A ) = ad - bc R × , then A M 2 ( R ) × . [ Hint:
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Unformatted text preview: Show that the usual formula yields the inverse of A .] c. Conclude that A ∈ M 2 ( R ) × if and only if det( A ) ∈ R × . Exercise 3. Given that Z × n = U ( n ) for n ≥ 2, prove that Z × n = Z n- { } if and only if n is prime. This completes the proof that Z n is a ﬁeld if and only if n is prime. Exercise 4. Let A = ² 1 3-2 4 ³ . Determine if A is a unit in M 2 ( R ) for the following choices of R . a. R = Q b. R = Z c. R = Z n (your answer will depend on n )...
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