Let e ,e2, ez denote, respectively, the following elementary row operations: "Interchange R, and R3," "Replace R, by 3R,," "Replace R, by 2R3 +R;" (a) Find the corresponding elementary matrices E , E2, Ey. (b) Find the inverse operations e-', e,', ez'; their corresponding elementary matrices E, E, E; and the relationship between them and Ej, E2, E3. (c) Describe the corresponding elementary column operations f1,f2 f3. (d) Find elementary matrices F1, F2, F3 corresponding to fi,f2,f3, and the relationship between them and E1, E2 , E3.

Let e ,e2, ez denote, respectively, the following elementary row operations: "Interchange R, and R3," "Replace R, by 3R,," "Replace R, by 2R3 +R;" (a) Find the corresponding elementary matrices E , E2, Ey. (b) Find the inverse operations e-', e,', ez'; their corresponding elementary matrices E, E, E; and the relationship between them and Ej, E2, E3. (c) Describe the corresponding elementary column operations f1,f2 f3. (d) Find elementary matrices F1, F2, F3 corresponding to fi,f2,f3, and the relationship between them and E1, E2 , E3.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.1: Definition Of A Group

Problem 36E: Consider the matrices R=[ 0110 ] H=[ 1001 ] V=[ 1001 ] D=[ 0110 ] T=[ 0110 ] in GL(2,), and let G={...

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Consider the matrices R=[ 0110 ] H=[ 1001 ] V=[ 1001 ] D=[ 0110 ] T=[ 0110 ] in GL(2,), and let G={ I2,R,R2,R3,H,D,V,T }. Given that G is a group of order 8 with respect to multiplication, write out a multiplication table for G. Sec. 3.3,22b,32b Find the center Z(G) for each of the following groups G. b. G={ I2,R,R2,R3,H,D,V,T } in Exercise 36 of section 3.1. Find the centralizer for each element a in each of the following groups. b. G={ I2,R,R2,R3,H,D,V,T } in Exercise 36 of section 3.1 Sec. 4.1,22 22. Find an isomorphism from the octic group D4 in Example 12 of this section to the group G={ I2,R,R2,R3,H,D,V,T } in Exercise 36 of Section 3.1. Sec. 4.6,14 14. Let G={ I2,R,R2,R3,H,D,V,T } be the multiplicative group of matrices in Exercise 36 of section 3.1, let G={ 1,1 } under multiplication, and define :GG by ([ abcd ])=adbc. Assume that is an epimorphism, and find the elements of K= ker . Write out the distinct elements of G/K. Let :G/KG be the isomorphism described in the proof of Theorem 4.27, and write out the values of .