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Lesson: Properties of Inverse Matrices

Sample Question Videos

Worksheet • 15 Questions • 1 Video

Q1:

Given that find ο€Ή 𝐴  𝑇 βˆ’ 1 .

  • A βŽ› ⎜ ⎜ ⎝ 7 5 8 βˆ’ 4 2 9 9 5 8 βˆ’ 1 2 9 ⎞ ⎟ ⎟ ⎠
  • B βŽ› ⎜ ⎜ ⎝ 7 5 8 9 5 8 βˆ’ 4 2 9 βˆ’ 1 2 9 ⎞ ⎟ ⎟ ⎠
  • C ο€Ό 7 βˆ’ 8 9 βˆ’ 2 
  • D βŽ› ⎜ ⎜ ⎝ βˆ’ 1 2 9 4 2 9 βˆ’ 9 5 8 7 5 8 ⎞ ⎟ ⎟ ⎠
  • E ο€Ό βˆ’ 2 8 βˆ’ 9 7 

Q2:

Consider the matrix 𝐴 = ο€Ό βˆ’ 3 1 βˆ’ 2 5  . Find ο€Ή 𝐴  βˆ’ 1 βˆ’ 1 .

  • A ο€Ό βˆ’ 3 1 βˆ’ 2 5 
  • B ο€Ό βˆ’ 5 2 βˆ’ 1 3 
  • C βŽ› ⎜ ⎜ ⎝ βˆ’ 5 1 3 1 1 3 βˆ’ 2 1 3 3 1 3 ⎞ ⎟ ⎟ ⎠
  • D ο€Ό 3 βˆ’ 1 2 βˆ’ 5 
  • E βŽ› ⎜ ⎜ ⎝ βˆ’ 3 1 3 2 1 3 βˆ’ 1 1 3 5 1 3 ⎞ ⎟ ⎟ ⎠

Q3:

Consider the matrix 𝐴 = ο€Ό βˆ’ 6 βˆ’ 1 2 0  . Find ο€Ή 𝐴  βˆ’ 1 βˆ’ 1 .

  • A ο€Ό βˆ’ 6 βˆ’ 1 2 0 
  • B ο€Ό 0 βˆ’ 2 1 6 
  • C  0 1 2 βˆ’ 1 βˆ’ 3 
  • D ο€Ό 6 1 βˆ’ 2 0 
  • E  3 1 βˆ’ 1 2 0 

Q4:

Consider the matrix 𝐴 = ο€Ό βˆ’ 5 βˆ’ 4 6 2  . Find ο€Ή 𝐴  βˆ’ 1 βˆ’ 1 .

  • A ο€Ό βˆ’ 5 βˆ’ 4 6 2 
  • B ο€Ό βˆ’ 2 βˆ’ 6 4 5 
  • C βŽ› ⎜ ⎜ ⎝ 1 7 2 7 βˆ’ 3 7 βˆ’ 5 1 4 ⎞ ⎟ ⎟ ⎠
  • D ο€Ό 5 4 βˆ’ 6 βˆ’ 2 
  • E βŽ› ⎜ ⎜ ⎝ 5 1 4 3 7 βˆ’ 2 7 βˆ’ 1 7 ⎞ ⎟ ⎟ ⎠

Q5:

Consider the matrix 𝐴 = ο€Ό βˆ’ 5 βˆ’ 1 3 βˆ’ 2  . Find ο€Ή 𝐴  βˆ’ 1 βˆ’ 1 .

  • A ο€Ό βˆ’ 5 βˆ’ 1 3 βˆ’ 2 
  • B ο€Ό 2 βˆ’ 3 1 5 
  • C βŽ› ⎜ ⎜ ⎝ βˆ’ 2 1 3 1 1 3 βˆ’ 3 1 3 βˆ’ 5 1 3 ⎞ ⎟ ⎟ ⎠
  • D ο€Ό 5 1 βˆ’ 3 2 
  • E βŽ› ⎜ ⎜ ⎝ 5 1 3 3 1 3 βˆ’ 1 1 3 2 1 3 ⎞ ⎟ ⎟ ⎠

Q6:

If 𝐴 is a matrix, which of the following is equal to ο€Ή 𝐴  βˆ’ 1 𝑑 ?

  • A ο€Ή 𝐴  𝑑 βˆ’ 1
  • B 𝐴 βˆ’ 1
  • C 𝐴 𝑑
  • D 𝐴 1 𝑑
  • E ο€Ή 𝐴  βˆ’ 1 1 𝑑

Q7:

Given that determine 𝐡 βˆ’ 1 .

  • A βŽ› ⎜ ⎜ ⎝ βˆ’ 1 6 1 6 1 2 βˆ’ 3 2 ⎞ ⎟ ⎟ ⎠
  • B ο€Ό βˆ’ 9 βˆ’ 1 βˆ’ 3 βˆ’ 1 
  • C ο€Ό 2 1 3 3 3 5 
  • D ο€Ό βˆ’ 1 1 3 βˆ’ 9 

Q8:

Given that find 𝐡 𝐴 βˆ’ 1 βˆ’ 1 .

  • A βŽ› ⎜ ⎜ ⎝ 1 1 2 6 7 2 6 8 1 3 9 2 6 ⎞ ⎟ ⎟ ⎠
  • B βŽ› ⎜ ⎜ ⎝ βˆ’ 9 2 6 βˆ’ 7 2 6 βˆ’ 8 1 3 βˆ’ 1 1 2 6 ⎞ ⎟ ⎟ ⎠
  • C βŽ› ⎜ ⎜ ⎝ 9 2 6 7 2 6 8 1 3 1 1 2 6 ⎞ ⎟ ⎟ ⎠
  • D βŽ› ⎜ ⎜ ⎝ βˆ’ 1 1 2 6 βˆ’ 7 2 6 βˆ’ 8 1 3 βˆ’ 9 2 6 ⎞ ⎟ ⎟ ⎠

Q9:

Suppose 𝐴 , 𝐡 , and 𝐢 are invertible 𝑛 Γ— 𝑛 matrices. Which of the following is false?

  • A 𝐴 𝐡 𝐢 βˆ’ 1 does not have full rank.
  • B ( 𝐴 𝐡 𝐢 ) = 𝐢 𝐡 𝐴 βˆ’ 1 βˆ’ 1 βˆ’ 1 βˆ’ 1
  • C d e t ( 𝐢 𝐴 𝐡 ) β‰  0
  • D 𝐴 𝐢 𝐡 βˆ’ 1 is invertible.

Q10:

If 𝐴 is a matrix, which of the following is equal to ο€Ή 𝐴  βˆ’ 1 2 ?

  • A ο€Ή 𝐴  2 βˆ’ 1
  • B 𝐴 2
  • C 𝐴 1 2
  • D ο€Ή 𝐴  βˆ’ 1 1 2

Q11:

If 𝐼 is the identity matrix, which of the following is equal to 𝐼   ?

  • A 𝐼
  • B βˆ’ 𝐼
  • C  1 1 1 1 
  • D  0 1 1 0 

Q12:

Given that find ο€Ή 𝐴  βˆ’ 1 βˆ’ 1 .

  • A ο€Ό βˆ’ 6 βˆ’ 2 1 βˆ’ 3 
  • B ο€Ό βˆ’ 3 1 βˆ’ 2 βˆ’ 6 
  • C ο€Ό βˆ’ 6 2 βˆ’ 1 βˆ’ 3 
  • D ο€Ό βˆ’ 3 βˆ’ 1 2 βˆ’ 6 

Q13:

Given that find ο€Ή 𝐴  βˆ’ 1 βˆ’ 1 .

  • A ο€Ό βˆ’ 7 βˆ’ 2 βˆ’ 1 6 
  • B ο€Ό 6 βˆ’ 1 βˆ’ 2 βˆ’ 7 
  • C ο€Ό βˆ’ 7 2 1 6 
  • D ο€Ό 6 1 2 βˆ’ 7 

Q14:

Given that find ο€Ή 𝐴  βˆ’ 1 βˆ’ 1 .

  • A ο€Ό βˆ’ 2 βˆ’ 5 4 βˆ’ 1 
  • B ο€Ό βˆ’ 1 4 βˆ’ 5 βˆ’ 2 
  • C ο€Ό βˆ’ 2 5 βˆ’ 4 βˆ’ 1 
  • D ο€Ό βˆ’ 1 βˆ’ 4 5 βˆ’ 2 

Q15:

If 𝐴 and 𝐡 are both nonsingular matrices, then what is the value of ( 𝐴 𝐡 ) βˆ’ 1 ?

  • A 𝐡 𝐴 βˆ’ 1 βˆ’ 1
  • B βˆ’ 𝐴 𝐡
  • C ( 𝐡 𝐴 ) βˆ’ 1
  • D 𝐴 𝐡 βˆ’ 1 βˆ’ 1
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