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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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 6 = 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 6 = 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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online