Question: Let R be a commutative ring with unity and M2(R) the ring of 2 2 matrices with entries from R. Show that A E M2(R)

Let R be a commutative ring with unity and M2(R) the ring of 2 2 matrices with entries from R. Show that A E M2(R) is a unit if and only if det (A) is a unit in Hint: There is a nice way to find the inverse of a 2 2 matrix. Hint 2: You may want to use that det(AB) = det(A) det(B) ) Find the units of M2(Z2)

Step by Step Solution

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock blur-text-image

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Students Have Also Explored These Related Databases Questions!

With reference to Exercise 12, suppose we drop the condition that T have no divisors of zero and just require that nonempty T not containing 0 be closed under multiplication. The attempt to enlarge R...

1. (10+5 pts) (a) Let / and J be two left ideals of R and let It J = {xty : re l, ye J}. Prove that I + J is a left ideal of R. (b) Formulate and prove an analogous statement for two-sided ideals. 2....

6. List all the polynomials of degree 2 in Zz[x]. Which of these are equal as functions from Z2 to Zz? 7. Find two distinct cubic polynomials over Z2 that determine the same function from Zz to Z2....

Questions of Abstract Algebra: th & W N = =t . Let H be a subgroup of G of index 2. Prove that H is a normal subgroup of G. . Find the order of the group of rigid motions of a cube. . Give the...

2. Let R be a commutative ring with identity and let u be a unit in R. Show that if a bu, then (a) = (b). = 3. For the following ideals I in the rings R, describe the quotient ring R/I by finding the...

Math 3310 - Algebraic Structures I 12/12/11 Final Ir Give precise denitions 4 of the following 7. (5 points each) 1 2 3 4 5 6 7 II. Work 3 of the following 5 problems. Determine if the statement is...

only with question 2 and 3. Abstract algebra (2) Let P be a set of prime numbers, not necessarily nite. Consider the set S of all rational numbers whose denominators in the reduced forms only have...

A. Answer the following questions: B. Q1. Write down the following set in rule method: C = {2,5,10,17,26,37,50} Q2. Construct a truth table for the proposition: (pq)v(~p-~q) Q3. Show that the...

if do - a, + a2 - . . . (-1)*a, is divisible by 11. 33. Show that the number 7, 176,825,942, 116,027,211 is divisible by 9 but not divisible by 11. 34. Show that the number 9,897,654,527,609,805 is...

1. Find an integer n that shows the rings Z, need not have the following properties that the ring of integers has. a) as = a implies a = 0 or a = 1 b) ab = 0 implies a = 0 or b = 0 c) ab = ac and a #...

Why may the financial information in an unadjusted trial balance not be up-to-date and complete?

Prime Contracting Services provides various services to government agencies under multi-year contracts. In 2006, the services primarily involved transportation of equipment and furniture. Beginning...

Skyrocket Company has a debt - equity ratio of . 7 4 . Return on assets is 9 . 6 percent, and total equity is $ 8 0 0 , 0 0 0 . What is the Net income?

Keep-or-Drop, Services, Qualitative Aspects Jem Dawson owns Jems Special Event Planning Service, a full-service event planner. Jem does much of the work herself and hires additional help as needed....

7. Senior management supports the career system.

10. Information about career plans and talent evaluation is accessible to all managers.

2. Discuss the implications that the career development model presented in this chapter may have for training and development activities.