2A3. Given the following matrices23[031]H[Bijl= 0 5 1 [Cijl= 1 0 2021243Demonstrate the equivalence of the following subscripted equations and the correspondingmatrix equations.[a](a) Dji = Bij [D] = [B]T,(b) b; = B¡¡a¡ [b] = [B][a],(c) cj = Bjai [c] = [B][a],(d) s = B¡a¡ª¡ s = [a]¹[B][a],(e) Dik = BijCjk [D] = [B][C],(f) Dik = BijCkj [D] = [B][C]T.

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Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.6: Rank Of A Matrix And Systems Of Linear Equations

Problem 77E: Let A and B be square matrices of order n satisfying, Ax=Bx for all x in all Rn. a Find the rank and...

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Let A,D, and P be nn matrices satisfying AP=PD. Assume that P is nonsingular and solve this for A. Must it be true that A=D?
Let A and B be square matrices of order n satisfying, Ax=Bx for all x in all Rn. a Find the rank and nullity of AB. b Show that matrices A and B must be identical.
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 .

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2A3. Given the following matrices 23 [031] H [Bijl= 0 5 1 [Cijl= 1 0 2 021 243 Demonstrate the equivalence of the following subscripted equations and the corresponding matrix equations. [a] (a) Dji = Bij [D] = [B]T, (b) b; = B¡¡a¡ [b] = [B][a], (c) c₁ = Bai [c] = [B][a],