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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)
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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Consider the following theorem. Properties of Transposes If 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 = A 2. (A + B)T = AT + BT 3. (CA)T = C(AT) 4. (AB)T = BTAT transpose of a transpose transpose of a sum transpose of a scalar multiple transpose of a product Let 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' = = ij Which 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)
Transcribed Image Text:Consider the following theorem. Properties of Transposes If 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 = A 2. (A + B)T = AT + BT 3. (CA)T = C(AT) 4. (AB)T = BTAT transpose of a transpose transpose of a sum transpose of a scalar multiple transpose of a product Let 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' = = ij Which 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 + ... j2 O a;b11 + a;b22 + ajbnn a nnbij + ... + ainbni * anpin + ... + a The entry in the ith row and the jth column of BTAT is which of the following? + ... + bnna ij O b11ªj + b22³ij O b1ajı + b2ªj2 O bja 11 + bja22 O bijaij + biza2i + bnjn + ... + bjann bina ni .. + bjna ni + b nj@in Select--- v the corresponding entry in B'A', the theorem has been proven. Since each entry in (AB)'
Transcribed Image Text: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 + ... j2 O a;b11 + a;b22 + ajbnn a nnbij + ... + ainbni * anpin + ... + a The entry in the ith row and the jth column of BTAT is which of the following? + ... + bnna ij O b11ªj + b22³ij O b1ajı + b2ªj2 O bja 11 + bja22 O bijaij + biza2i + bnjn + ... + bjann bina ni .. + bjna ni + b nj@in Select--- v the corresponding entry in B'A', the theorem has been proven. Since each entry in (AB)'
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