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Worksheet: Power of a Matrix

Q1:

Which of the following represents a matrix 𝐴 such that 𝐴 = 𝐼 2 , yet 𝐴 β‰  𝐼 and 𝐴 β‰  βˆ’ 𝐼 ?

  • A 𝐴 =  1 1 1 1 1 1 1 1 1 
  • B 𝐴 =  1 1 1 0 1 0 1 1 0 
  • C 𝐴 =  1 0 0 0 1 0 1 1 0 
  • D 𝐴 =  1 0 0 0 βˆ’ 1 0 0 0 1 
  • E 𝐴 =  1 0 0 0 1 0 1 1 1 

Q2:

Consider the matrix

Find 𝐴 2 .

  • A 𝐴 =  2 2 4 2 0 2 4 2 0  2
  • B 𝐴 =  1 1 4 1 0 1 4 1 0  2
  • C 𝐴 =  6 3 3 3 1 2 3 1 5  2
  • D 𝐴 =  6 3 3 3 2 2 3 2 5  2
  • E 𝐴 =  1 1 4 1 0 1 4 1 0  2

Find 𝐴 3 .

  • A 𝐴 =  1 5 9 1 5 9 5 8 1 5 8 8  3
  • B 𝐴 =  1 5 9 1 5 9 5 8 1 5 8 8  3
  • C 𝐴 =  3 3 6 3 0 3 6 3 0  3
  • D 𝐴 =  1 1 8 1 0 1 8 1 0  3
  • E 𝐴 =  3 3 6 3 0 3 6 3 0  3

Q3:

Consider the matrices and . What is ?

  • A
  • B
  • C
  • D

Q4:

Which of the following statements is true for all 𝑛 Γ— 𝑛 matrices 𝐴 and 𝐡 ?

  • A ( 𝐴 + 𝐡 ) ( 𝐴 βˆ’ 𝐡 ) = 𝐴 βˆ’ 𝐡 2 2
  • B ( 𝐴 𝐡 ) = 𝐴 𝐡 2 2 2
  • C ( 𝐴 + 𝐡 ) = 𝐴 + 2 𝐴 𝐡 + 𝐡 2 2 2
  • D 𝐴 𝐡 = 𝐴 ( 𝐴 𝐡 ) 𝐡 2 2
  • E ( 𝐴 βˆ’ 𝐡 ) = 𝐴 βˆ’ 2 𝐴 𝐡 + 𝐡 2 2 2

Q5:

For write as a multiple of .

  • A
  • B
  • C
  • D
  • E

Q6:

Find and .

  • A ,
  • B ,
  • C ,
  • D ,
  • E ,

Q7:

Given that is a zero matrix, find .

  • A
  • B
  • C
  • D
  • E

Q8:

Which of the following statements is true for all 𝑛 Γ— 𝑛 matrices 𝐴 and 𝐡 ?

  • A ( 𝐴 βˆ’ 𝐡 ) = 𝐴 βˆ’ 2 𝐴 𝐡 + 𝐡 2 2 2
  • B ( 𝐴 + 𝐡 ) = 𝐴 + 2 𝐴 𝐡 + 𝐡 2 2 2
  • C ( 𝐴 + 𝐡 ) = 𝐴 + 3 𝐴 𝐡 + 3 𝐴 𝐡 + 𝐡 3 3 2 2 3
  • D ( 𝐴 + 𝐡 ) = 𝐴 + 𝐴 𝐡 + 𝐡 𝐴 + 𝐡 2 2 2
  • E ( 𝐴 + 𝐡 ) = 𝐴 + 2 𝐡 𝐴 + 𝐡 2 2 2

Q9:

Given that the eigenvalues of the nondefective 𝑛 Γ— 𝑛 matrix 𝐴 are 1 and βˆ’ 1 , find 𝐴 1 2 .

  • A 𝐴 = 1 2 𝐼 1 2
  • B 𝐴 = βˆ’ 𝐼 1 2
  • C 𝐴 = βˆ’ 1 2 𝐼 1 2
  • D 𝐴 = 𝐼 1 2
  • E 𝐴 = 𝐼 1 2 1 2