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.

  • Aπ‘Ž=23√5, 𝑏=βˆ’53√5, 𝑐=23, 𝑑=βˆ’1√5
  • Bπ‘Ž=βˆ’23√5, 𝑏=βˆ’53√5, 𝑐=23, 𝑑=βˆ’1√5
  • Cπ‘Ž=βˆ’23√5, 𝑏=53√5, 𝑐=23, 𝑑=1√5
  • Dπ‘Ž=23√5, 𝑏=53√5, 𝑐=βˆ’23, 𝑑=1√5
  • Eπ‘Ž=23√5, 𝑏=βˆ’53√5, 𝑐=23, 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.

  • Aπ‘Ž=βˆ’βˆš22, 𝑏=13√2, 𝑐=13, 𝑑=43√2
  • Bπ‘Ž=√22, 𝑏=13√2, 𝑐=βˆ’13, 𝑑=βˆ’43√2
  • Cπ‘Ž=βˆ’βˆš22, 𝑏=13√2, 𝑐=βˆ’13, 𝑑=βˆ’43√2
  • Dπ‘Ž=√22, 𝑏=13√2, 𝑐=13, 𝑑=43√2
  • Eπ‘Ž=βˆ’βˆš22, 𝑏=βˆ’13√2, 𝑐=13, 𝑑=43√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.

  • Aπ‘Ž=βˆ’1√6, 𝑏=1√3, 𝑐=0, 𝑑=1√3
  • Bπ‘Ž=βˆ’1√6, 𝑏=βˆ’1√3, 𝑐=0, 𝑑=βˆ’1√3
  • Cπ‘Ž=1√6, 𝑏=βˆ’1√3, 𝑐=0, 𝑑=βˆ’1√3
  • Dπ‘Ž=βˆ’1√6, 𝑏=βˆ’1√3, 𝑐=0, 𝑑=1√3
  • Eπ‘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.

  • AβŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£βˆ’1√21√6√1261√2βˆ’1√6βˆ’βˆš1260√63√126⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦
  • BβŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£βˆ’1√21√6√1261√2βˆ’1√6βˆ’βˆš1261√63√126⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦
  • CβŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£βˆ’1√21√6√1261√21√6√1261√63βˆ’βˆš126⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦
  • DβŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£βˆ’1√21√6√1261√2βˆ’1√6√1260√63βˆ’βˆš126⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦
  • EβŽ‘βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ’βŽ£βˆ’1√21√6√1261√21√6√1260√63βˆ’βˆš126⎀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?

  • Adet(𝐴)=1√2 or det(𝐴)=βˆ’1√2
  • Bdet(𝐴)=0
  • Cdet(𝐴)=1 or det(𝐴)=βˆ’1
  • Ddet(𝐴)=√2 or det(𝐴)=βˆ’βˆš2
  • Edet(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?

  • A𝐴=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₯⎦.

  • A𝐴 is an orthogonal matrix.
  • B𝐴 is not an orthogonal matrix.

Q13:

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

  • AThe matrix is orthogonal.
  • BThe matrix is not orthogonal.

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