hw06 - a,b,c,d Z and ad-bc 6 = 0 . Prove or disprove: G is...

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Modern Algebra 1 Homework 3.2 Spring 2010 Due February 3 Exercise 6. Given a 2 × 2 matrix A = ± a b c d ² , recall that it’s determinant is det A = ad - bc . The transpose of A is the matrix A T = ± a c b d ² . Prove the following properties of the determinant and the transpose. You can assume that the matrices involved all have real entires. a. If A and B are both 2 × 2 matrices then det AB = det A det B . b. If A and B are both 2 × 2 matrices, then ( AB ) T = B T A T . c. If det A 6 = 0 (so that A - 1 exists) then ( A T ) - 1 = ( A - 1 ) T . Exercise 7. Let G = ³± a b c d ²´
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Unformatted text preview: a,b,c,d Z and ad-bc 6 = 0 . Prove or disprove: G is a group under matrix multiplication. Exercise 8. Prove that O 2 ( R ) = { A M 2 ( R ) | AA T = I } is a subgroup of GL 2 ( R ). O 2 ( R ) is called the orthogonal group and its elements are the orthogonal matrices . Exercise 9. Let G be a group, H G and x G . Prove that xHx-1 = { xyx-1 | y H } is a subgroup of G ....
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