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).
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).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text: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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