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
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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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Question
![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)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fccfefc58-5b13-48ad-9d77-94c8c32f66a8%2F0562dd7e-d9a9-41a8-9e70-7c516b0da5a3%2F13sbkhe_processed.jpeg&w=3840&q=75)
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)'](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fccfefc58-5b13-48ad-9d77-94c8c32f66a8%2F0562dd7e-d9a9-41a8-9e70-7c516b0da5a3%2Fzzn57ga_processed.jpeg&w=3840&q=75)
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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