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Worksheet: Orthogonal Matrices

Q1:

Is the matrix orthogonal?

  • AYes
  • BNo

Q2:

Fill in the missing entries to make the matrix orthogonal.

  • A 1 2 1 6 1 2 6 1 2 1 6 1 2 6 0 6 3 1 2 6
  • B 1 2 1 6 1 2 6 1 2 1 6 1 2 6 1 6 3 1 2 6
  • C 1 2 1 6 1 2 6 1 2 1 6 1 2 6 1 6 3 1 2 6
  • D 1 2 1 6 1 2 6 1 2 1 6 1 2 6 0 6 3 1 2 6
  • E 1 2 1 6 1 2 6 1 2 1 6 1 2 6 0 6 3 1 2 6

Q3:

Given that the matrix is orthogonal, find the values of 𝑎 , 𝑏 , 𝑐 , and 𝑑 .

  • A 𝑎 = 1 6 , 𝑏 = 1 3 , 𝑐 = 0 , 𝑑 = 1 3
  • B 𝑎 = 1 6 , 𝑏 = 1 3 , 𝑐 = 0 , 𝑑 = 1 3
  • C 𝑎 = 1 6 , 𝑏 = 1 3 , 𝑐 = 0 , 𝑑 = 1 3
  • D 𝑎 = 1 6 , 𝑏 = 1 3 , 𝑐 = 0 , 𝑑 = 1 3
  • E 𝑎 = 1 6 , 𝑏 = 1 3 , 𝑐 = 0 , 𝑑 = 1 3

Q4:

Is the given matrix orthogonal?

  • A yes
  • B no

Q5:

Is the matrix orthogonal?

  • A Yes
  • B No

Q6:

Given that the matrix is orthogonal, find the values of 𝑎 , 𝑏 , 𝑐 , and 𝑑 .

  • A 𝑎 = 2 2 , 𝑏 = 1 3 2 , 𝑐 = 1 3 , 𝑑 = 4 3 2
  • B 𝑎 = 2 2 , 𝑏 = 1 3 2 , 𝑐 = 1 3 , 𝑑 = 4 3 2
  • C 𝑎 = 2 2 , 𝑏 = 1 3 2 , 𝑐 = 1 3 , 𝑑 = 4 3 2
  • D 𝑎 = 2 2 , 𝑏 = 1 3 2 , 𝑐 = 1 3 , 𝑑 = 4 3 2
  • E 𝑎 = 2 2 , 𝑏 = 1 3 2 , 𝑐 = 1 3 , 𝑑 = 4 3 2

Q7:

Given that the matrix is orthogonal, find the values of 𝑎 , 𝑏 , 𝑐 , and 𝑑 .

  • A 𝑎 = 2 3 5 , 𝑏 = 5 3 5 , 𝑐 = 2 3 , 𝑑 = 1 5
  • B 𝑎 = 2 3 5 , 𝑏 = 5 3 5 , 𝑐 = 2 3 , 𝑑 = 1 5
  • C 𝑎 = 2 3 5 , 𝑏 = 5 3 5 , 𝑐 = 2 3 , 𝑑 = 1 5
  • D 𝑎 = 2 3 5 , 𝑏 = 5 3 5 , 𝑐 = 2 3 , 𝑑 = 1 5
  • E 𝑎 = 2 3 5 , 𝑏 = 5 3 5 , 𝑐 = 2 3 , 𝑑 = 1 5

Q8:

Is the matrix orthogonal?

  • A No
  • B Yes

Q9:

A matrix is said to be orthogonal if 𝐴 𝐴 = 𝐼 𝑇 . Thus, the inverse of an orthogonal matrix is just its transpose. What are the possible values of d e t ( 𝐴 ) if 𝐴 is an orthogonal matrix?

  • A d e t ( 𝐴 ) = 1 2 or d e t ( 𝐴 ) = 1 2
  • B d e t ( 𝐴 ) = 0
  • C d e t ( 𝐴 ) = 2
  • D d e t ( 𝐴 ) = 1 or d e t ( 𝐴 ) = 1
  • E d e t ( 𝐴 ) = 2 or d e t ( 𝐴 ) = 2

Q10:

True or false: the matrix is orthogonal?

  • ATrue
  • B False