Worksheet: Power of a Matrix

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In this worksheet, we will practice using the matrix multiplication to determine the square and cube of a square matrix.

Q1:

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

Q2:

Given that the eigenvalues of the nondefective 𝑛 × 𝑛 matrix 𝐴 are 1 and − 1 , find 𝐴 1 2 .

  • A 𝐴 = 1 2 𝐼 1 2
  • B 𝐴 = − 𝐼 1 2
  • C 𝐴 = − 1 2 𝐼 1 2
  • D 𝐴 = 𝐼 1 2
  • E 𝐴 = 𝐼 1 2 1 2

Q3:

Find ⎛ ⎜ ⎜ ⎝ 3 2 1 − 1 2 0 ⎞ ⎟ ⎟ ⎠ 3 5 and l i m 𝑛 → ∞ 𝑛 ⎛ ⎜ ⎜ ⎝ 3 2 1 − 1 2 0 ⎞ ⎟ ⎟ ⎠ .

  • A 1 0 0 1 2 3 5 , 1 0 0 0
  • B ⎛ ⎜ ⎜ ⎝ 2 + 1 2 2 + 2 2 1 2 + 1 2 2 + 1 ⎞ ⎟ ⎟ ⎠ 3 5 3 5 3 5 3 5 , 2 2 1 1
  • C 1 0 0 1 2 3 5 , 1 0 0 1
  • D ⎛ ⎜ ⎜ ⎝ 2 − 1 2 2 − 2 2 1 2 − 1 2 2 − 1 ⎞ ⎟ ⎟ ⎠ 3 5 3 5 3 5 3 5 , 2 2 − 1 − 1
  • E ⎛ ⎜ ⎜ ⎝ 2 1 2 − 1 − 1 1 + 1 2 ⎞ ⎟ ⎟ ⎠ 3 5 3 5 , 2 − 1 − 1 1

Q4:

Given that 𝑂 is a 3 × 3 zero matrix, find 𝑂 .

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

Q5:

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

  • A
  • B
  • C
  • D
  • E

Q6:

Which of the following statements is true for all 𝑛 × 𝑛 matrices 𝐴 and 𝐵 ?

  • A ( 𝐴 + 𝐵 ) ( 𝐴 − 𝐵 ) = 𝐴 − 𝐵 2 2
  • B ( 𝐴 𝐵 ) = 𝐴 𝐵 2 2 2
  • C ( 𝐴 + 𝐵 ) = 𝐴 + 2 𝐴 𝐵 + 𝐵 2 2 2
  • D 𝐴 𝐵 = 𝐴 ( 𝐴 𝐵 ) 𝐵 2 2
  • E ( 𝐴 − 𝐵 ) = 𝐴 − 2 𝐴 𝐵 + 𝐵 2 2 2

Q7:

For write 𝐴 2 as a multiple of 𝐴 .

  • A 𝐴
  • B 4 𝐴
  • C − 4 𝐴
  • D − 𝐴
  • E 2 𝐴

Q8:

Which of the following statements is true for all 𝑛 × 𝑛 matrices 𝐴 and 𝐵 ?

  • A ( 𝐴 − 𝐵 ) = 𝐴 − 2 𝐴 𝐵 + 𝐵 2 2 2
  • B ( 𝐴 + 𝐵 ) = 𝐴 + 2 𝐴 𝐵 + 𝐵 2 2 2
  • C ( 𝐴 + 𝐵 ) = 𝐴 + 3 𝐴 𝐵 + 3 𝐴 𝐵 + 𝐵 3 3 2 2 3
  • D ( 𝐴 + 𝐵 ) = 𝐴 + 𝐴 𝐵 + 𝐵 𝐴 + 𝐵 2 2 2
  • E ( 𝐴 + 𝐵 ) = 𝐴 + 2 𝐵 𝐴 + 𝐵 2 2 2

Q9:

Consider What is 𝑋 − 𝑌 ?

  • A − 2 5 − 3 0 5 7 − 3 3
  • B − 2 5 − 1 5 4 2 − 3 3
  • C − 2 5 5 7 − 3 0 − 3 3
  • D − 2 5 4 2 − 1 5 − 3 3

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