Worksheet: Properties of Inverse Matrices

In this worksheet, we will practice using some properties of matrix inverse.

Q1:

Given that find 𝐴 𝑇 1 .

  • A 2 8 9 7
  • B 7 8 9 2
  • C 1 2 9 4 2 9 9 5 8 7 5 8
  • D 7 5 8 4 2 9 9 5 8 1 2 9
  • E 7 5 8 9 5 8 4 2 9 1 2 9

Q2:

If 𝐴 is a matrix, which of the following is equal to 𝐴 1 2 ?

  • A 𝐴 1 1 2
  • B 𝐴 2
  • C 𝐴 1 2
  • D 𝐴 2 1

Q3:

Consider the matrix 𝐴 = 3 1 2 5 . Find 𝐴 1 1 .

  • A 3 1 3 2 1 3 1 1 3 5 1 3
  • B 5 1 3 1 1 3 2 1 3 3 1 3
  • C 3 1 2 5
  • D 3 1 2 5
  • E 5 2 1 3

Q4:

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

  • A 𝐴 𝐵 1 1
  • B 𝐴 𝐵
  • C ( 𝐵 𝐴 ) 1
  • D 𝐵 𝐴 1 1

Q5:

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

  • A 0 1 1 0
  • B 𝐼
  • C 1 1 1 1
  • D 𝐼

Q6:

Given that find 𝐴 1 1 .

  • A 3 1 2 6
  • B 3 1 2 6
  • C 6 2 1 3
  • D 6 2 1 3

Q7:

Given that determine 𝐵 1 .

  • A 1 1 3 9
  • B 9 1 3 1
  • C 2 1 3 3 3 5
  • D 1 6 1 6 1 2 3 2

Q8:

Given that find 𝐵 𝐴 1 1 .

  • A 1 1 2 6 7 2 6 8 1 3 9 2 6
  • B 9 2 6 7 2 6 8 1 3 1 1 2 6
  • C 9 2 6 7 2 6 8 1 3 1 1 2 6
  • D 1 1 2 6 7 2 6 8 1 3 9 2 6

Q9:

If 𝐴 is a matrix, which of the following is equal to 𝐴 1 𝑡 ?

  • A 𝐴 1 1 𝑡
  • B 𝐴 𝑡
  • C 𝐴 1 𝑡
  • D 𝐴 𝑡 1
  • E 𝐴 1

Q10:

Suppose 𝐴 , 𝐵 , and 𝐶 are invertible 𝑛 × 𝑛 matrices. Which of the following is false?

  • A 𝐴 𝐶 𝐵 is invertible.
  • B ( 𝐴 𝐵 𝐶 ) = 𝐶 𝐵 𝐴
  • C d e t ( 𝐶 𝐴 𝐵 ) 0
  • D 𝐴 𝐵 𝐶 does not have full rank.

Q11:

Consider the matrix 𝐴 = 6 1 2 0 . Find 𝐴 1 1 .

  • A 3 1 1 2 0
  • B 0 1 2 1 3
  • C 6 1 2 0
  • D 6 1 2 0
  • E 0 2 1 6

Q12:

Consider the matrix 𝐴 = 5 4 6 2 . Find 𝐴 1 1 .

  • A 5 1 4 3 7 2 7 1 7
  • B 1 7 2 7 3 7 5 1 4
  • C 5 4 6 2
  • D 5 4 6 2
  • E 2 6 4 5

Q13:

Consider the matrix 𝐴 = 5 1 3 2 . Find 𝐴 1 1 .

  • A 5 1 3 3 1 3 1 1 3 2 1 3
  • B 2 1 3 1 1 3 3 1 3 5 1 3
  • C 5 1 3 2
  • D 5 1 3 2
  • E 2 3 1 5

Q14:

Given that find 𝐴 1 1 .

  • A 6 1 2 7
  • B 6 1 2 7
  • C 7 2 1 6
  • D 7 2 1 6

Q15:

Given that find 𝐴 1 1 .

  • A 1 4 5 2
  • B 1 4 5 2
  • C 2 5 4 1
  • D 2 5 4 1

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