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 𝐴=112101210.

Find 𝐴.

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

Find 𝐴.

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

Q2:

Which of the following represents a matrix 𝐴 such that 𝐴=𝐼, yet 𝐴≠𝐼 and 𝐴≠−𝐼?

  • A 𝐴 = 1 0 0 0 − 1 0 0 0 1
  • B 𝐴 = 1 1 1 1 1 1 1 1 1
  • C 𝐴 = 1 1 1 0 1 0 1 1 0
  • D 𝐴 = 1 0 0 0 1 0 1 1 0
  • E 𝐴 = 1 0 0 0 1 0 1 1 1

Q3:

For 𝐴=4−54−5, write 𝐴 as a multiple of 𝐴.

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

Q4:

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

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

Q5:

Consider 𝑋=−3−35−6,𝑌=136−6. What is 𝑋−𝑌?

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

Q6:

Given that 𝐴=−61−55, find 𝐴.

  • A 3 1 − 1 5 2 0
  • B 3 7 3 5 3 5 5 0
  • C 3 1 5 − 1 2 0
  • D 6 1 − 3 1 − 3 1 2 6

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