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Aula: Matrizes Ortogonais

Atividade • 9 Questões

Q1:

A matriz dada Γ© ortogonal? 𝐴 = 1 3  1 βˆ’ 2 2 2 βˆ’ 1 βˆ’ 2 2 2 1 

  • A nΓ£o
  • B sim

Q2:

Preencha as entradas em falta para tornar a matriz ⎑ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎣ βˆ’ 1 √ 2 1 √ 6 √ 1 2 6 1 √ 2 β‹― β‹― β‹― √ 6 3 β‹― ⎀ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ ⎦ ortogonal.

  • A ⎑ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎣ βˆ’ 1 √ 2 1 √ 6 √ 1 2 6 1 √ 2 1 √ 6 √ 1 2 6 0 √ 6 3 βˆ’ √ 1 2 6 ⎀ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ ⎦
  • B ⎑ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎣ βˆ’ 1 √ 2 1 √ 6 √ 1 2 6 1 √ 2 βˆ’ 1 √ 6 √ 1 2 6 0 √ 6 3 βˆ’ √ 1 2 6 ⎀ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ ⎦
  • C ⎑ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎣ βˆ’ 1 √ 2 1 √ 6 √ 1 2 6 1 √ 2 1 √ 6 √ 1 2 6 1 √ 6 3 βˆ’ √ 1 2 6 ⎀ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ ⎦
  • D ⎑ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎣ βˆ’ 1 √ 2 1 √ 6 √ 1 2 6 1 √ 2 βˆ’ 1 √ 6 βˆ’ √ 1 2 6 1 √ 6 3 √ 1 2 6 ⎀ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ ⎦
  • E ⎑ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎣ βˆ’ 1 √ 2 1 √ 6 √ 1 2 6 1 √ 2 βˆ’ 1 √ 6 βˆ’ √ 1 2 6 0 √ 6 3 √ 1 2 6 ⎀ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ ⎦

Q3:

Sabendo que a matriz ⎑ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎣ βˆ’ 1 √ 2 βˆ’ 1 √ 6 1 √ 3 1 √ 2 π‘Ž 𝑏 𝑐 √ 6 3 𝑑 ⎀ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ ⎦ Γ© ortogonal, determine os valores de π‘Ž , 𝑏 , 𝑐 e 𝑑 .

  • A π‘Ž = βˆ’ 1 √ 6 , 𝑏 = 1 √ 3 , 𝑐 = 0 , 𝑑 = 1 √ 3
  • B π‘Ž = βˆ’ 1 √ 6 , 𝑏 = βˆ’ 1 √ 3 , 𝑐 = 0 , 𝑑 = 1 √ 3
  • C π‘Ž = 1 √ 6 , 𝑏 = 1 √ 3 , 𝑐 = 0 , 𝑑 = 1 √ 3
  • D π‘Ž = 1 √ 6 , 𝑏 = βˆ’ 1 √ 3 , 𝑐 = 0 , 𝑑 = βˆ’ 1 √ 3
  • E π‘Ž = βˆ’ 1 √ 6 , 𝑏 = βˆ’ 1 √ 3 , 𝑐 = 0 , 𝑑 = βˆ’ 1 √ 3

Q4:

A matriz dada Γ© ortogonal? 𝐴 =  πœƒ πœƒ βˆ’ πœƒ πœƒ  c o s s e n s e n c o s

  • A nΓ£o
  • B sim

Q5:

A matriz dada Γ© ortogonal? 𝐴 =  πœƒ πœƒ 0 βˆ’ πœƒ πœƒ 0 0 0 1 ο₯ c o s s e n s e n c o s

  • A nΓ£o
  • B sim

Q6:

Sabendo que a matriz ⎑ ⎒ ⎒ ⎒ ⎒ ⎣ 2 3 √ 2 2 √ 2 6 2 3 π‘Ž 𝑏 𝑐 0 𝑑 ⎀ βŽ₯ βŽ₯ βŽ₯ βŽ₯ ⎦ Γ© ortogonal, determine os valores de π‘Ž , 𝑏 , 𝑐 e 𝑑 .

  • A π‘Ž = βˆ’ √ 2 2 , 𝑏 = 1 3 √ 2 , 𝑐 = βˆ’ 1 3 , 𝑑 = βˆ’ 4 3 √ 2
  • B π‘Ž = βˆ’ √ 2 2 , 𝑏 = βˆ’ 1 3 √ 2 , 𝑐 = 1 3 , 𝑑 = 4 3 √ 2
  • C π‘Ž = √ 2 2 , 𝑏 = 1 3 √ 2 , 𝑐 = βˆ’ 1 3 , 𝑑 = βˆ’ 4 3 √ 2
  • D π‘Ž = √ 2 2 , 𝑏 = 1 3 √ 2 , 𝑐 = 1 3 , 𝑑 = 4 3 √ 2
  • E π‘Ž = βˆ’ √ 2 2 , 𝑏 = 1 3 √ 2 , 𝑐 = 1 3 , 𝑑 = 4 3 √ 2

Q7:

Dado que a matriz ⎑ ⎒ ⎒ ⎒ ⎒ ⎒ ⎒ ⎣ 1 3 βˆ’ 2 √ 5 π‘Ž 2 3 0 𝑏 𝑐 𝑑 4 √ 5 1 5 ⎀ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ βŽ₯ ⎦ Γ© ortogonal, encontre os valores de π‘Ž , 𝑏 , 𝑐 , e 𝑑 .

  • A π‘Ž = 2 3 √ 5 , 𝑏 = βˆ’ 5 3 √ 5 , 𝑐 = 2 3 , 𝑑 = 1 √ 5
  • B π‘Ž = βˆ’ 2 3 √ 5 , 𝑏 = βˆ’ 5 3 √ 5 , 𝑐 = 2 3 , 𝑑 = βˆ’ 1 √ 5
  • C π‘Ž = βˆ’ 2 3 √ 5 , 𝑏 = 5 3 √ 5 , 𝑐 = 2 3 , 𝑑 = 1 √ 5
  • D π‘Ž = 2 3 √ 5 , 𝑏 = βˆ’ 5 3 √ 5 , 𝑐 = 2 3 , 𝑑 = βˆ’ 1 √ 5
  • E π‘Ž = 2 3 √ 5 , 𝑏 = 5 3 √ 5 , 𝑐 = βˆ’ 2 3 , 𝑑 = 1 √ 5

Q8:

A matriz dada é ortogonal? 𝐴 =  1 2 2 2 1 2 2 2 1 

  • A sim
  • B nΓ£o

Q9:

Uma matriz diz-se ortogonal se 𝐴 𝐴 = 𝐼 𝑇 . Portanto, a inversa de uma matriz ortogonal Γ© apenas a sua transposta. Quais sΓ£o os valores possΓ­veis de d e t ( 𝐴 ) se 𝐴 for uma matriz ortogonal?

  • A d e t ( 𝐴 ) = 1 ou d e t ( 𝐴 ) = βˆ’ 1
  • B d e t ( 𝐴 ) = √ 2 ou d e t ( 𝐴 ) = βˆ’ √ 2
  • C d e t ( 𝐴 ) = 0
  • D d e t ( 𝐴 ) = 2
  • E d e t ( 𝐴 ) = 1 √ 2 ou d e t ( 𝐴 ) = βˆ’ 1 √ 2
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