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Chapter 5 Solutions
Elements Of Modern Algebra
- Let R be the set of all matrices of the form [abba], where a and b are real numbers. Assume that R is a commutative ring with unity with respect to matrix addition and multiplication. Answer the following questions and give a reason for any negative answers. Is 12 an integral domain? Is R a field?arrow_forwardConsider the matrices below. X=[1201],Y=[1032],Z=[3412],W=[3241] Find scalars a,b, and c such that W=aX+bY+cZ. Show that there do not exist scalars a and b such that Z=aX+bY. Show that if aX+bY+cZ=0, then a=b=c=0.arrow_forward28. a. Show that the set is a ring with respect to matrix addition and multiplication. b. Is commutative? c. does have a unity? d. Decide whether or not the set is an ideal of and justify your answer.arrow_forward
- 15. Let and be elements of a ring. Prove that the equation has a unique solution.arrow_forward1. (12%) Let R be the set of all 2 x 2 matrices of the form where x is an integer. Prove that R is a ring with respect to matrix addition and multiplication.arrow_forwardJ Let Cring complex numbers and M ring from the set of all - matrices of the from (a b) for each a, b € R. e a show that C = M.arrow_forward
- Let M2x2 (R) be the set of all 2 × 2 commutative and invertible matrices with real entries. Does M2x2 (R) form a field? Prove your assertion. Determine whether M2x2 (R) form an ordered field or not? If your answer is 'no'; can we define an order such that M2x2 (R) form an order field? Prove your assertion.arrow_forward{(6 ):a e Z, b, e e Q. Together with matrix addition and multiplica- a E! cE Q 1 Let R = tion, show that Ris a ring. Determine the units and left zero divisors of R.arrow_forwardа Prove that the ring of matrices bEQ is a field. 2b a, aarrow_forward
- Let R be the ring of all 2 x 2 matrices over Z w.r.t. matrix addition and matrix multiplication. Then show that i) B= {(6 ): a, b e 2} is a right ideal of R, but not left ideal of R. %3D 00arrow_forwardLet B1,B2, . . . , Bk be square matrices with entries in the same field, and let A = B1⊕B2⊕·· ·⊕Bk. Prove that the characteristic polynomial of A is the product of the characteristic polynomials of the Bi’s.arrow_forwardLet A = [1 -1 -1 -1 1 -1 -1 -1 1]. Find a diagonalization if possible.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage LearningElementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning