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Lesson: Power of a Matrix

In this lesson, we will learn how to use the matrix multiplication to determine the square and cube of a square matrix.

Worksheet • 9 Questions

Q1:

Which of the following represents a matrix such that , yet and ?

  • A
  • B
  • C
  • D
  • E

Q2:

Consider the matrix

Find 𝐴 2 .

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

Find 𝐴 3 .

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

Q3:

Consider What is 𝑋 βˆ’ π‘Œ   ?

  • A  βˆ’ 2 5 4 2 βˆ’ 1 5 βˆ’ 3 3 
  • B  βˆ’ 2 5 βˆ’ 1 5 4 2 βˆ’ 3 3 
  • C  βˆ’ 2 5 5 7 βˆ’ 3 0 βˆ’ 3 3 
  • D  βˆ’ 2 5 βˆ’ 3 0 5 7 βˆ’ 3 3 

Q4:

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

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

Q5:

For write 𝐴 2 as a multiple of 𝐴 .

  • A βˆ’ 𝐴
  • B 2 𝐴
  • C 4 𝐴
  • D βˆ’ 4 𝐴
  • E 𝐴

Q6:

Find βŽ› ⎜ ⎜ ⎝ 3 2 1 βˆ’ 1 2 0 ⎞ ⎟ ⎟ ⎠ 3 5 and l i m 𝑛 β†’ ∞ 𝑛 βŽ› ⎜ ⎜ ⎝ 3 2 1 βˆ’ 1 2 0 ⎞ ⎟ ⎟ ⎠ .

  • A βŽ› ⎜ ⎜ ⎝ 2 βˆ’ 1 2 2 βˆ’ 2 2 1 2 βˆ’ 1 2 2 βˆ’ 1 ⎞ ⎟ ⎟ ⎠ 3 5 3 5 3 5 3 5 , ο€Ό 2 2 βˆ’ 1 βˆ’ 1 
  • B βŽ› ⎜ ⎜ ⎝ 2 1 2 βˆ’ 1 βˆ’ 1 1 + 1 2 ⎞ ⎟ ⎟ ⎠ 3 5 3 5 , ο€Ό 2 βˆ’ 1 βˆ’ 1 1 
  • C βŽ› ⎜ ⎜ ⎝ 2 + 1 2 2 + 2 2 1 2 + 1 2 2 + 1 ⎞ ⎟ ⎟ ⎠ 3 5 3 5 3 5 3 5 , ο€Ό 2 2 1 1 
  • D  1 0 0 1 2  3 5 , ο€Ό 1 0 0 1 
  • E  1 0 0 1 2  3 5 , ο€Ό 1 0 0 0 

Q7:

Given that 𝑂 is a 3 Γ— 3 zero matrix, find 𝑂 𝑑 .

  • A 𝑂
  • B  1 0 0 0 1 0 0 0 1 
  • C 𝐼
  • D  1 1 1 1 1 1 1 1 1 
  • E  0 1 1 1 0 1 1 1 0 

Q8:

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

  • A ( 𝐴 + 𝐡 ) = 𝐴 + 𝐴 𝐡 + 𝐡 𝐴 + 𝐡 2 2 2
  • B ( 𝐴 + 𝐡 ) = 𝐴 + 2 𝐡 𝐴 + 𝐡 2 2 2
  • C ( 𝐴 + 𝐡 ) = 𝐴 + 2 𝐴 𝐡 + 𝐡 2 2 2
  • D ( 𝐴 + 𝐡 ) = 𝐴 + 3 𝐴 𝐡 + 3 𝐴 𝐡 + 𝐡 3 3 2 2 3
  • E ( 𝐴 βˆ’ 𝐡 ) = 𝐴 βˆ’ 2 𝐴 𝐡 + 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 = 𝐼 1 2
  • B 𝐴 = 𝐼 1 2 1 2
  • C 𝐴 = βˆ’ 𝐼 1 2
  • D 𝐴 = βˆ’ 1 2 𝐼 1 2
  • E 𝐴 = 1 2 𝐼 1 2
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