Consider the following theorem.Properties of TransposesIf A and B are matrices (with sizes such that the matrix operations are defined) and c is a scalar, then the properties below are true.1. (AT)T = A2. (A + B)T = AT + BT3. (CA)T = C(AT)4. (AB)T = BTATtranspose of a transposetranspose of a sumtranspose of a scalar multipletranspose of a productLet A =and B =Which of the following proves Property 2 from the above theorem?O (A + B)T = ([] + [»])" - [-, + bg]° = [e» + bg] = [)· Þ)-O (A + B)T - ([,] + [»J])' - [-]"-= AT + BT= A' + BT= AT + BT+++==== A' + BT+=%3D+ou T- () · Þ.])' - [», + +]' - [Þ, + •g] - [Þ.] · ] - [P.] • [•»] - ^T + s"O (4 + n)" - ( + [*J) - [-, - " - [v*•] = [•] + [-)] - [-] + [•J = a' +n'==ijWhich of the following proves Property 3 from the above theorem?O (CA)T = (c{=])" = [ca,]" = [ca,] = c[a,] = caT)O (CA)T = (c[a])" - [cn] - [-J] = <[•J] = ««a">==O (A)T = ([L) = [[-] ]' = [e,] = [•][•] = clat)O (CA)T = (cla,)" = [c + a;]" = [c + a„] = c[a] - c(AT) O (CA)' = (cl*]) - [ca - [ca - cl27] - C(A')O (aA)T = (clej) = [a)" - () = cle] = ca")O (c4)T = (c/2,)" = [ + a.]" - [- + -] = c[=,] = caT)O (CA)T - ([L) - [[G]]" - [] - GL-da")O (CA)T = (c[e,)" = [ + a,] - [-+ •;] = c[•] = ca")====Prove Property 4 from the above theorem.The entry in the ith row and the jth column of (AB)T is which of the following?+ a.+ a in nj+ ...+ a nPjn+ ...j2O a;b11 + a;b22+ ajbnna nnbij+ ... + ainbni* anpin+ ... + aThe entry in the ith row and the jth column of BTAT is which of the following?+ ... + bnna ijO b11ªj + b22³ijO b1ajı + b2ªj2O bja 11 + bja22O bijaij + biza2i+ bnjn+ ... + bjannbina ni.. + bjna ni+ bnj@inSelect---v the corresponding entry in B'A', the theorem has been proven.Since each entry in (AB)'

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O (CA)' = (cl*]) - [ca - [ca - cl27] - C(A') O (CA)T = (cle])" = [ca]' - [] = c[ed] = cta™) O (c4)T = (c/2,)" = [ + a.]" - [- + -] = c[=,] = caT) O (A)T - ([L) - [[G]]" - [] - G]-da") O (CA)T = (c[,)" = [ + a,] - [-+ •;] = c[•] = ca") = = = Prove Property 4 from the above theorem. The entry in the ith row and the jth column of (AB)T is which of the following? a in nj + a. + ... O a, bj + azb;2 O a,b11 + a ,b22 + ... + a nº jn + ajbnn a nnbij + ... + ajnbni + anjbin

O (CA)' = (cl*]) - [ca - [ca - cl27] - C(A') O (CA)T = (cle])" = [ca]' - [] = c[ed] = cta™) O (c4)T = (c/2,)" = [ + a.]" - [- + -] = c[=,] = caT) O (A)T - ([L) - [[G]]" - [] - G]-da") O (CA)T = (c[,)" = [ + a,] - [-+ •;] = c[•] = ca") = = = Prove Property 4 from the above theorem. The entry in the ith row and the jth column of (AB)T is which of the following? a in nj + a. + ... O a, bj + azb;2 O a,b11 + a ,b22 + ... + a nº jn + ajbnn a nnbij + ... + ajnbni + anjbin

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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