2) Consider the set SL(n, R) consisting of n x n matrices with real entries having de-terminant equal to 1. Prove that SL(n, R) is a group under the operation of matrixmultiplication (it is referred to as the Special Linear Group).

To prove that SL(n,R) the set of n×n real matrices with determinant 1, forms a group under…

Answered: 2) Consider the set SL(n, R) consisting…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.4: The Singular Value Decomposition

Problem 33EQ

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Let SL₂(R)= {A|A is a 2x 2 matrix with det(A)=1}. We will show that SL₂(R) is a group with matrix multiplication: (a) Let A,BE SL₂(R). Find det(AB). (b) IS AB E SL₂(R) ? O No, SL₂(R) is not closed under multiplication. Yes, SL₂(R) is closed under multiplication. (c) I= We already know that matrix multiplication is associative and that 0 is a multiplicative identity for matrix multiplication. Find det(I). 01 (d) Is IE SL₂(R) ? O No, SL₂(R) does not contain an identity element. O Yes, SL₂(R) contains an identity element. (e) Let AE SL₂(R). Find det(A-¹). (f) Is A-¹ ESL₂(R) ? O Yes, every element of SL2(R) has an inverse in SL₂(R). O No, not every element of SL₂(R) has an inverse in SL₂(R).

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2) Consider the set SL(n, R) consisting of n x n matrices with real entries having de- terminant equal to 1. Prove that SL(n, R) is a group under the operation of matrix multiplication (it is referred to as the Special Linear Group).