Worksheet: Orthogonal Matrices

In this worksheet, we will practice determining whether a matrix is orthogonal and finding its inverse if it is.

Q1:

Is the matrix 𝐴=131βˆ’222βˆ’1βˆ’2221 orthogonal?

  • ANo
  • BYes

Q2:

Is the given matrix orthogonal? 𝐴=ο”πœƒπœƒβˆ’πœƒπœƒο cossinsincos

  • AYes
  • BNo

Q3:

Is the matrix 𝐴=122212221 orthogonal?

  • ANo
  • BYes

Q4:

Is the matrix 𝐴=ο™πœƒπœƒ0βˆ’πœƒπœƒ0001ο₯cossinsincos orthogonal?

  • AYes
  • BNo

Q5:

Given that the matrix ⎑⎒⎒⎒⎒⎒⎒⎣13βˆ’2√5π‘Ž230𝑏𝑐𝑑4√515⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦ is orthogonal, find the values of π‘Ž, 𝑏, 𝑐, and 𝑑.

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

Q6:

Given that the matrix ⎑⎒⎒⎒⎒⎣23√22√2623π‘Žπ‘π‘0π‘‘βŽ€βŽ₯βŽ₯βŽ₯βŽ₯⎦ is orthogonal, find the values of π‘Ž, 𝑏, 𝑐, and 𝑑.

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

Q7:

Given that the matrix βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£βˆ’1√2βˆ’1√61√31√2π‘Žπ‘π‘βˆš63π‘‘βŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦ is orthogonal, find the values of π‘Ž, 𝑏, 𝑐, and 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 = βˆ’ 1 √ 3
  • C π‘Ž = 1 √ 6 , 𝑏 = βˆ’ 1 √ 3 , 𝑐 = 0 , 𝑑 = βˆ’ 1 √ 3
  • D π‘Ž = βˆ’ 1 √ 6 , 𝑏 = βˆ’ 1 √ 3 , 𝑐 = 0 , 𝑑 = 1 √ 3
  • E π‘Ž = 1 √ 6 , 𝑏 = 1 √ 3 , 𝑐 = 0 , 𝑑 = 1 √ 3

Q8:

Fill in the missing entries to make the matrix βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£βˆ’1√21√6√1261√2β‹―β‹―β‹―βˆš63β‹―βŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦ orthogonal.

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

Q9:

A matrix is said to be orthogonal if 𝐴𝐴=𝐼. Thus, the inverse of an orthogonal matrix is just its transpose. What are the possible values of det(𝐴) if 𝐴 is an orthogonal matrix?

  • A d e t ( 𝐴 ) = 1 √ 2 or det(𝐴)=βˆ’1√2
  • B d e t ( 𝐴 ) = 0
  • C d e t ( 𝐴 ) = 1 or det(𝐴)=βˆ’1
  • D d e t ( 𝐴 ) = √ 2 or det(𝐴)=βˆ’βˆš2
  • E d e t ( 𝐴 ) = 2

Q10:

True or false: the matrix 01βˆ’10 is orthogonal?

  • AFalse
  • BTrue

Q11:

If 𝐴 is an π‘šΓ—π‘š orthogonal matrix, which of the following may not be true about 𝐴?

  • A 𝐴 = 𝐴    .
  • BThe rows of 𝐴 form an orthonormal basis for ℝ.
  • CThe determinant of 𝐴 equals 1.
  • DThe columns of 𝐴 form an orthonormal basis for ℝ.
  • EThe nullity of 𝐴 is 0.

Q12:

Determine whether the following matrix is orthogonal: 𝐴=⎑⎒⎒⎒⎣√32βˆ’1212√32⎀βŽ₯βŽ₯βŽ₯⎦.

  • A 𝐴 is an orthogonal matrix.
  • B 𝐴 is not an orthogonal matrix.

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