Worksheet: Inverse of a 2x2 Matrix

In this worksheet, we will practice checking whether a 2x2 matrix has an inverse and then finding its inverse, if possible.

Q1:

Find the inverse of the matrix 𝐴=3316.

  • A𝐴=1216βˆ’3βˆ’13
  • B𝐴=115ο”βˆ’313βˆ’6
  • C𝐴=1156313
  • D𝐴=121ο”βˆ’313βˆ’6
  • E𝐴=1156βˆ’3βˆ’13

Q2:

Find the inverse of the matrix 𝐴=4βˆ’237.

  • A𝐴=134ο”βˆ’43βˆ’2βˆ’7
  • B𝐴=1347βˆ’234
  • C𝐴=122ο”βˆ’43βˆ’2βˆ’7
  • D𝐴=12272βˆ’34
  • E𝐴=13472βˆ’34

Q3:

Is the matrix 51βˆ’15 invertible?

  • AYes
  • BNo

Q4:

Is the matrix 31βˆ’31 invertible?

  • AYes
  • BNo

Q5:

Find the set of real values of π‘Ž for which 𝐴=ο”π‘Ž251π‘Žο  has a multiplicative inverse.

  • Aβ„βˆ’{5}
  • Bℝ
  • Cβ„βˆ’{25,1}
  • D{5,βˆ’5}
  • Eβ„βˆ’{5,βˆ’5}

Q6:

Given that 𝐴=π‘₯βˆ’π‘₯𝑦0𝑦, find 𝐴.

  • A⎑⎒⎒⎒⎣1π‘₯101π‘¦βŽ€βŽ₯βŽ₯βŽ₯⎦
  • B⎑⎒⎒⎒⎣1π‘₯βˆ’101π‘¦βŽ€βŽ₯βŽ₯βŽ₯⎦
  • CβŽ‘βŽ’βŽ’βŽ’βŽ£βˆ’1π‘₯01βˆ’1π‘¦βŽ€βŽ₯βŽ₯βŽ₯⎦
  • D⎑⎒⎒⎒⎣1π‘₯011π‘¦βŽ€βŽ₯βŽ₯βŽ₯⎦

Q7:

Are the matrices 1234,⎑⎒⎒⎣1121314⎀βŽ₯βŽ₯⎦ multiplicative inverses of each other?

  • AYes
  • BNo

Q8:

Find the inverse of the following matrix. 𝐴=ο”βˆ’11βˆ’18

  • A𝐴=19ο”βˆ’8βˆ’1βˆ’1βˆ’1
  • B𝐴=17ο”βˆ’81βˆ’11
  • C𝐴=178βˆ’11βˆ’1
  • D𝐴=19ο”βˆ’81βˆ’11
  • E𝐴=198βˆ’11βˆ’1

Q9:

Find the multiplicative inverse of the matrix 𝐴=ο•βˆ’4βˆ’1035, if possible.

  • A𝐴has no multiplicative inverse.
  • Bο•βˆ’410βˆ’35
  • C⎑⎒⎒⎒⎣121βˆ’310βˆ’25⎀βŽ₯βŽ₯βŽ₯⎦
  • D510βˆ’3βˆ’4
  • E5βˆ’103βˆ’4

Q10:

Given that the matrix 71βˆ’7π‘Žο  is invertible, what must be true of π‘Ž.

  • Aπ‘Žβ‰ βˆ’1
  • Bπ‘Žβ‰ 7
  • Cπ‘Žβ‰ 1
  • Dπ‘Žβ‰ βˆ’7
  • Eπ‘Žβ‰ 0

Q11:

Find the multiplicative inverse of 𝐴=ο”βˆ’48βˆ’1224, if possible.

  • A𝐴has no multiplicative inverse.
  • B2412βˆ’8βˆ’4
  • Cβˆ’11922412βˆ’8βˆ’4
  • Dβˆ’119224βˆ’812βˆ’4

Q12:

Given 𝐴=4βˆ’7βˆ’3βˆ’4, find its multiplicative inverse if possible.

  • Aβˆ’137ο”βˆ’4734
  • Bβˆ’137ο”βˆ’4374
  • Cο”βˆ’4734
  • D𝐴 has no multiplicative inverse.

Q13:

Find the multiplicative inverse of 690069.

  • Aο”βˆ’6900βˆ’69
  • B⎑⎒⎒⎣16900169⎀βŽ₯βŽ₯⎦
  • CβŽ‘βŽ’βŽ’βŽ£βˆ’16900βˆ’169⎀βŽ₯βŽ₯⎦
  • D6900βˆ’69

Q14:

Given that 𝐡=ο”βˆ’2βˆ’5βˆ’6βˆ’10,𝐴×𝐡=𝐼, find the matrix 𝐴.

  • A⎑⎒⎒⎣1βˆ’12βˆ’3515⎀βŽ₯βŽ₯⎦
  • B⎑⎒⎒⎣1512351⎀βŽ₯βŽ₯⎦
  • Cο”βˆ’1056βˆ’2
  • Dο”βˆ’10βˆ’5βˆ’6βˆ’2

Q15:

Find the multiplicative inverse of ο•πœƒπœƒ1πœƒο‘.sectansec

  • Aο•πœƒβˆ’1βˆ’πœƒπœƒο‘sectansec
  • Bο•πœƒβˆ’πœƒβˆ’1πœƒο‘sectansec
  • Cο•πœƒπœƒπœƒβˆ’1tansecsec
  • Dο•πœƒπœƒ1πœƒο‘sectansec

Q16:

Find the multiplicative inverse of ο”πœƒβˆ’πœƒπœƒπœƒο .sincoscossin

  • Aο”πœƒβˆ’πœƒπœƒπœƒο cossinsincos
  • Bο”βˆ’πœƒπœƒβˆ’πœƒβˆ’πœƒο cossinsincos
  • Cο”πœƒβˆ’πœƒπœƒπœƒο sincoscossin
  • Dο”πœƒπœƒβˆ’πœƒπœƒο sincoscossin

Q17:

Find the multiplicative inverse of the matrix 𝐴=3722, if possible.

  • A𝐴has no multiplicative inverse.
  • B3βˆ’7βˆ’22
  • CβŽ‘βŽ’βŽ’βŽ’βŽ£βˆ’147814βˆ’38⎀βŽ₯βŽ₯βŽ₯⎦
  • D2βˆ’7βˆ’23
  • E2723

Q18:

Find the multiplicative inverse of the matrix 𝐴=5βˆ’27βˆ’2, if possible.

  • A𝐴has no multiplicative inverse.
  • B52βˆ’7βˆ’2
  • CβŽ‘βŽ’βŽ’βŽ’βŽ£βˆ’1212βˆ’7454⎀βŽ₯βŽ₯βŽ₯⎦
  • Dο•βˆ’22βˆ’75
  • Eο•βˆ’2βˆ’275

Q19:

Is there any value of 𝑑 for which the matrix 1000π‘‘βˆ’π‘‘0𝑑𝑑ο₯cossinsincos has no inverse?

  • Ayes, when 𝑑=πœ‹3
  • Bno
  • Cyes, when 𝑑=πœ‹2
  • Dyes, when 𝑑=πœ‹
  • Eyes, when 𝑑=πœ‹6

Q20:

Is there any value of 𝑑 for which the matrix οšπ‘’π‘‘π‘‘π‘’π‘‘π‘‘π‘’π‘‘π‘‘ο¦οοοcoshsinhsinhcoshcoshsinh has no inverse?

  • Ayes, when 𝑑=0
  • Byes, when 𝑑=1
  • Cno
  • Dyes, the matrix has no inverse for all values of 𝑑
  • Eyes, when 𝑑=πœ‹

Q21:

Under what condition on π‘˜ is the matrix ο”βˆ’6βˆ’π‘˜βˆ’61211βˆ’π‘˜ο  invertible?

  • Aπ‘˜=2 or π‘˜=βˆ’3
  • Bπ‘˜ is any real number.
  • Cπ‘˜β‰ 2 and π‘˜β‰ 3
  • Dπ‘˜=2 or π‘˜=3
  • Eπ‘˜β‰ βˆ’6 and π‘˜β‰ 11

Q22:

Find the set of real values of π‘Ž for which ο•π‘Žπ‘–βˆ’π‘–1 has a multiplicative inverse, where 𝑖=βˆ’1.

  • Aβ„βˆ’{0}
  • Bℝ
  • Cβ„βˆ’{1}
  • Dβ„βˆ’{βˆ’1}

Q23:

Find the value of π‘₯ that makes the matrix π‘₯βˆ’8βˆ’5βˆ’1 singular.

Q24:

Find the set of real values of π‘Ž for which 𝐴=ο”π‘Žβˆ’7410π‘Žβˆ’1 has a multiplicative inverse.

  • Aβ„βˆ’{4,10}
  • Bβ„βˆ’{11,βˆ’3}
  • C{11,βˆ’3}
  • Dℝ
  • Eβ„βˆ’{βˆ’7,βˆ’1}

Q25:

Under what condition on π‘Ž and 𝑏 is the matrix ο”π‘Ž2𝑏5 invertible?

  • A5𝑏≠2π‘Ž
  • Bπ‘Žβ‰ π‘
  • C2𝑏≠5π‘Ž
  • D2𝑏=5π‘Ž
  • E2π‘Ž+5𝑏≠0

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