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# aug30 - Math 3124 Tuesday August 30 August 30 Ungraded...

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Math 3124 Tuesday, August 30 August 30 Ungraded Homework Problem 5.16 on page 34 Let G denote the set of all 2 × 2 real matrices A with 0 = det ( A ) Q . Prove that G is a group with respect to matrix multiplication. (You may assume that matrix multiplication is associative. But check closure (so matrix multiplication is an operation on G ), and the existence of an identity element and inverse elements very carefully.) Is G abelian? Let I denote the identity 2 × 2 matrix (1’s on the main diagonal and 0’s elsewhere). We will want to use the well-known property that det ( AB ) = det ( A ) det ( B ) for any 2 × 2 matrices A , B . If A , B G , then AB is also matrix with real entries. Since det ( AB ) = det ( A ) det ( B ) and the product of two nonzero rational numbers is a nonzero rational number, we see that 0 = det ( AB ) Q and we deduce that AB G . Thus we have closure and matrix multiplication is indeed an operation on G . Next matrix multiplication is associative (we are allowed to assume this). Furthermore I G and is the identity for

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aug30 - Math 3124 Tuesday August 30 August 30 Ungraded...

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