Consider the set of matrices {[1] (a) Prove that if a GL(Z), then there exists...

 

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Consider the set of matrices {[1] (a) Prove that if a € GL₂(Z), then there exists y E GL2(Z) such that [1 GL₂(Z) = a, b, c, d e Z and ad-bc= te=1} xy=yx = 1, where I = (b) Prove that if ₁,2 € GL2₂(Z), then 12 € GL₂(Z). [Remark: this shows that GL2(Z) is a subgroup of GL₂(R).] Consider the set of matrices {[1] (a) Prove that if a € GL₂(Z), then there exists y E GL2(Z) such that [1 GL₂(Z) = a, b, c, d e Z and ad-bc= te=1} xy=yx = 1, where I = (b) Prove that if ₁,2 € GL2₂(Z), then 12 € GL₂(Z). [Remark: this shows that GL2(Z) is a subgroup of GL₂(R).]

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