2. Define= {(9₂2) = 0,²€c}: a, Bec};(i) Prove MH is a subring of Mat2 (C) with usual matrix addition and multiplication.(ii) Prove it is a division ring.(iii) Prove it is non-commutative.(iv) Prove that the matrix product in MH reflects the product of quaternions described in class.(v) Find a linearly independent set of generators of MH as an R-vector space.MH

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Answered: 2. Define = {(9₂2) = 0,²€c} : a, Bec};…

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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1-What is the definition of the characteristic polynomial of a square matrix A? 2-
QI/ 1- Write the multiplication table of the ring (Zs, +5,'s). 2- Is H = (0,2} subring of the ring (Z4,+4i4) or not, why? %3D Q2/ Let (M2x2(R), +,) the ring of all 2x2 matrices. Show that if T, T2 are ideal or not: (a) T = {(" ) : a, b,c € Z} (b) T; = ((, ) : a, b e R} -) Q3/ 1- Let f: Z, → Z10 define by f(x) = 2x+2. Is fhomomorphism or not, explain your answer. 2- Let f be an isomorphism from the ring (R, +;) to the ring (R', +','). If (I, +;) is an ideal of (R', +',').Prove (f-(1), +,) is an ideal of (R, +;).
6. Prove that the Rotation Matrix R corresponding to quaternion q = (a, b, c, d) is given by 1. Ꭱ . = [2(a² +6²) - 1 2(bc + ad) 2(bd - ac) 2(bc - ad) 2(a² + c²) - 1 2(cd + ab) 2(bd + ac) 2(cd - ab) 2(a²+d²)-1]
3) Let Z, be ring integers mode 5 and f(x) € Zs(x) such that f(x) = x² + 2x + 1 then a) f(x) has only one root. b) f(x) has only two roots. c) f(x) irreducible polynomials.. d) No one of above. 4) Let R be a ring and F-M₂(R) be ring of 2 x 2 matrices over R under standard addition and multiplication of matrices then: a) F is a commutative ring but not field. b) F is a division ring and not field. (c) If F an integral domain and F is not field. d) No one of above.
Show that the set of all matrices of order 2×2 of the form [10], where a, b, c are integers is a subring of the ring M of all matrices of order 2 × 2 with integral elements.
1 2-4 -1 -1 5 2 7-3 a) A = Calculate the inverse of the matrix using the Cramer Methool b) write the characteristic polynomial of the giuen matiix using the Cayley Homilten Theorem.
To avoid any confusion, unless specified otherwise, vector spaces are complex vector spaces, inner products are complex inner-products, and matrices are complex matrices. True or False:
3: Which of the following sets are closed under scalar multiplication? (1) The set of all vectors in R of the form (a, b, 9ab). (i1) The set of all polynomials a- bx+ cx in P, such that ac =D0. (iii) The set of all matrices in Mn such that A = A (A) (1) and (ii) only (B) (1) and (iii) only (C) (ii) and (iii) only (D) (1) only (E) (ii) only (F) none of them (G) (iii) only (H) all of them
Suppose that β and γ are ordered bases for an n-dimensional real [complex] inner product space V. Prove that if Q is an orthogonal [unitary] n ×n matrix that changes γ-coordinates into β-coordinates, then β is orthonormal if and only if γ is orthonormal.
Regarded in M3(F5), where F5 is the field of five elements, put the matrix [0 2 4] 420 003 A = into canonical form for equivalence using only row operations. (Hint: you may wish to use that 2-¹ = 3, 3-¹ = 2, 4¯¹ = 4 when working in F5.)
Please give as much detail as possible
Let M2x2 (R) be the set of all matrices with dimension 2 × 2, whose entries are all real numbers. Prove that (M2x2(R), +, x) is a ring, if + and x are the addition and multiplication operations on (2×2)-dimension matrices, respectively.

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