Worksheet: Inverse of a 2x2 Matrix

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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.

  • A 𝐴 = 1 2 1 6 − 3 − 1 3
  • B 𝐴 = 1 1 5 − 3 1 3 − 6
  • C 𝐴 = 1 1 5 6 3 1 3
  • D 𝐴 = 1 2 1 − 3 1 3 − 6
  • E 𝐴 = 1 1 5 6 − 3 − 1 3

Q2:

Find the inverse of the matrix 𝐴=4−237.

  • A 𝐴 = 1 3 4 − 4 3 − 2 − 7
  • B 𝐴 = 1 3 4 7 − 2 3 4
  • C 𝐴 = 1 2 2 − 4 3 − 2 − 7
  • D 𝐴 = 1 2 2 7 2 − 3 4
  • E 𝐴 = 1 3 4 7 2 − 3 4

Q3:

Is the matrix 51−15 invertible?

  • AYes
  • BNo

Q4:

Is the matrix 31−31 invertible?

  • AYes
  • BNo

Q5:

Find the set of real values of 𝑎 for which 𝐴=𝑎251𝑎 has a multiplicative inverse.

  • A ℝ − { 5 }
  • B ℝ
  • C ℝ − { 2 5 , 1 }
  • D { 5 , − 5 }
  • E ℝ − { 5 , − 5 }

Q6:

Given that 𝐴=𝑥−𝑥𝑦0𝑦, find 𝐴.

  • A ⎡ ⎢ ⎢ ⎢ ⎣ 1 𝑥 1 0 1 𝑦 ⎤ ⎥ ⎥ ⎥ ⎦
  • B ⎡ ⎢ ⎢ ⎢ ⎣ 1 𝑥 − 1 0 1 𝑦 ⎤ ⎥ ⎥ ⎥ ⎦
  • C ⎡ ⎢ ⎢ ⎢ ⎣ − 1 𝑥 0 1 − 1 𝑦 ⎤ ⎥ ⎥ ⎥ ⎦
  • D ⎡ ⎢ ⎢ ⎢ ⎣ 1 𝑥 0 1 1 𝑦 ⎤ ⎥ ⎥ ⎥ ⎦

Q7:

Are the matrices 1234,⎡⎢⎢⎣1121314⎤⎥⎥⎦ multiplicative inverses of each other?

  • AYes
  • BNo

Q8:

Find the inverse of the following matrix. 𝐴=−11−18

  • A 𝐴 = 1 9 − 8 − 1 − 1 − 1
  • B 𝐴 = 1 7 − 8 1 − 1 1
  • C 𝐴 = 1 7 8 − 1 1 − 1
  • D 𝐴 = 1 9 − 8 1 − 1 1
  • E 𝐴 = 1 9 8 − 1 1 − 1

Q9:

Find the multiplicative inverse of the matrix 𝐴=−4−1035, if possible.

  • A 𝐴 has no multiplicative inverse.
  • B − 4 1 0 − 3 5
  • C ⎡ ⎢ ⎢ ⎢ ⎣ 1 2 1 − 3 1 0 − 2 5 ⎤ ⎥ ⎥ ⎥ ⎦
  • D 5 1 0 − 3 − 4
  • E 5 − 1 0 3 − 4

Q10:

Given that the matrix 71−7𝑎 is invertible, what must be true of 𝑎.

  • A 𝑎 ≠ − 1
  • B 𝑎 ≠ 7
  • C 𝑎 ≠ 1
  • D 𝑎 ≠ − 7
  • E 𝑎 ≠ 0

Q11:

Find the multiplicative inverse of 𝐴=−48−1224, if possible.

  • A 𝐴 has no multiplicative inverse.
  • B 2 4 1 2 − 8 − 4
  • C − 1 1 9 2 2 4 1 2 − 8 − 4
  • D − 1 1 9 2 2 4 − 8 1 2 − 4

Q12:

Given 𝐴=4−7−3−4, find its multiplicative inverse if possible.

  • A − 1 3 7 − 4 7 3 4
  • B − 1 3 7 − 4 3 7 4
  • C − 4 7 3 4
  • D 𝐴 has no multiplicative inverse.

Q13:

Find the multiplicative inverse of 690069.

  • A − 6 9 0 0 − 6 9
  • B ⎡ ⎢ ⎢ ⎣ 1 6 9 0 0 1 6 9 ⎤ ⎥ ⎥ ⎦
  • C ⎡ ⎢ ⎢ ⎣ − 1 6 9 0 0 − 1 6 9 ⎤ ⎥ ⎥ ⎦
  • D 6 9 0 0 − 6 9

Q14:

Given that 𝐵=−2−5−6−10,𝐴×𝐵=𝐼, find the matrix 𝐴.

  • A ⎡ ⎢ ⎢ ⎣ 1 − 1 2 − 3 5 1 5 ⎤ ⎥ ⎥ ⎦
  • B ⎡ ⎢ ⎢ ⎣ 1 5 1 2 3 5 1 ⎤ ⎥ ⎥ ⎦
  • C − 1 0 5 6 − 2
  • D − 1 0 − 5 − 6 − 2

Q15:

Find the multiplicative inverse of 𝜃𝜃1𝜃.sectansec

  • A 𝜃 − 1 − 𝜃 𝜃 s e c t a n s e c
  • B 𝜃 − 𝜃 − 1 𝜃 s e c t a n s e c
  • C 𝜃 𝜃 𝜃 − 1 t a n s e c s e c
  • D 𝜃 𝜃 1 𝜃 s e c t a n s e c

Q16:

Find the multiplicative inverse of 𝜃−𝜃𝜃𝜃.sincoscossin

  • A 𝜃 − 𝜃 𝜃 𝜃 c o s s i n s i n c o s
  • B − 𝜃 𝜃 − 𝜃 − 𝜃 c o s s i n s i n c o s
  • C 𝜃 − 𝜃 𝜃 𝜃 s i n c o s c o s s i n
  • D 𝜃 𝜃 − 𝜃 𝜃 s i n c o s c o s s i n

Q17:

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

  • A 𝐴 has no multiplicative inverse.
  • B 3 − 7 − 2 2
  • C ⎡ ⎢ ⎢ ⎢ ⎣ − 1 4 7 8 1 4 − 3 8 ⎤ ⎥ ⎥ ⎥ ⎦
  • D 2 − 7 − 2 3
  • E 2 7 2 3

Q18:

Find the multiplicative inverse of the matrix 𝐴=5−27−2, if possible.

  • A 𝐴 has no multiplicative inverse.
  • B 5 2 − 7 − 2
  • C ⎡ ⎢ ⎢ ⎢ ⎣ − 1 2 1 2 − 7 4 5 4 ⎤ ⎥ ⎥ ⎥ ⎦
  • D − 2 2 − 7 5
  • E − 2 − 2 7 5

Q19:

Is there any value of 𝑡 for which the matrix 1000𝑡−𝑡0𝑡𝑡cossinsincos has no inverse?

  • Ayes, when 𝑡=𝜋3
  • Bno
  • Cyes, when 𝑡=𝜋2
  • Dyes, when 𝑡=𝜋
  • Eyes, when 𝑡=𝜋6

Q20:

Is there any value of 𝑡 for which the matrix 𝑒𝑡𝑡𝑒𝑡𝑡𝑒𝑡𝑡coshsinhsinhcoshcoshsinh has no inverse?

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

Q21:

Under what condition on 𝑘 is the matrix −6−𝑘−61211−𝑘 invertible?

  • A 𝑘 = 2 or 𝑘=−3
  • B 𝑘 is any real number.
  • C 𝑘 ≠ 2 and 𝑘≠3
  • D 𝑘 = 2 or 𝑘=3
  • E 𝑘 ≠ − 6 and 𝑘≠11

Q22:

Find the set of real values of 𝑎 for which 𝑎𝑖−𝑖1 has a multiplicative inverse, where 𝑖=−1.

  • A ℝ − { 0 }
  • B ℝ
  • C ℝ − { 1 }
  • D ℝ − { − 1 }

Q23:

Find the value of 𝑥 that makes the matrix 𝑥−8−5−1 singular.

  • A40
  • B − 1 4 0
  • C − 4 0
  • D 1 4 0

Q24:

Find the set of real values of 𝑎 for which 𝐴=𝑎−7410𝑎−1 has a multiplicative inverse.

  • A ℝ − { 4 , 1 0 }
  • B ℝ − { 1 1 , − 3 }
  • C { 1 1 , − 3 }
  • D ℝ
  • E ℝ − { − 7 , − 1 }

Q25:

Under what condition on 𝑎 and 𝑏 is the matrix 𝑎2𝑏5 invertible?

  • A 5 𝑏 ≠ 2 𝑎
  • B 𝑎 ≠ 𝑏
  • C 2 𝑏 ≠ 5 𝑎
  • D 2 𝑏 = 5 𝑎
  • E 2 𝑎 + 5 𝑏 ≠ 0

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