Worksheet: Cayley–Hamilton Theorem

In this worksheet, we will practice using the Cayley–Hamilton theorem to solve problems.

Q1:

Using the Cayley-Hamilton theorem, find 𝐴 for the matrix 𝐴=3βˆ’154 if possible.

  • A𝐴=11741βˆ’53
  • B𝐴=41βˆ’53
  • C𝐴=1741βˆ’53
  • D𝐴=117ο”βˆ’4βˆ’15βˆ’3
  • E𝐴=117101βˆ’511

Q2:

Let 𝐴 be an 𝑛×𝑛 matrix with characteristic polynomial 𝑝. Which of the following is true?

  • A𝑝(𝐴)=0
  • B𝑝(𝐴)=𝐼
  • C𝑝(𝐴) is invertible.
  • D𝑝(𝐴)=𝐴

Q3:

Using the Cayley-Hamilton theorem, find 𝐴, if possible, for the matrix 𝐴=234βˆ’556789.

  • A𝐴=145ο˜βˆ’35βˆ’287βˆ’10βˆ’32βˆ’75525
  • B𝐴=1453βˆ’52βˆ’87103275βˆ’5βˆ’25
  • CThe matrix has no inverse.
  • D𝐴=1453βˆ’8775βˆ’510βˆ’5232βˆ’25
  • E𝐴=15673βˆ’52βˆ’87103275βˆ’5βˆ’25

Q4:

Consider the matrices 𝐴=12βˆ’3βˆ’4,𝐼=1001.

Find 𝐴.

  • A𝐴=14916
  • B𝐴=14916
  • C𝐴=7βˆ’6βˆ’922
  • D𝐴=ο”βˆ’5βˆ’6910
  • E𝐴=7101522

Find 𝐴+3𝐴+2𝐼.

  • A𝐴+3𝐴+2𝐼=64612
  • B𝐴+3𝐴+2𝐼=0000
  • C𝐴+3𝐴+2𝐼=0000
  • D𝐴+3𝐴+2𝐼=41606
  • E𝐴+3𝐴+2𝐼=120012

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