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.

  • A 𝐴 = 1 1 7  4 1 βˆ’ 5 3   
  • B 𝐴 =  4 1 βˆ’ 5 3   
  • C 𝐴 = 1 7  4 1 βˆ’ 5 3   
  • D 𝐴 = 1 1 7  βˆ’ 4 βˆ’ 1 5 βˆ’ 3   
  • E 𝐴 = 1 1 7  1 0 1 βˆ’ 5 1 1   

Q2:

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

  • A 𝑝 ( 𝐴 ) = 0
  • B 𝑝 ( 𝐴 ) = 𝐼
  • C 𝑝 ( 𝐴 ) is invertible.
  • D 𝑝 ( 𝐴 ) = 𝐴

Q3:

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

  • A 𝐴 = 1 4 5  βˆ’ 3 5 βˆ’ 2 8 7 βˆ’ 1 0 βˆ’ 3 2 βˆ’ 7 5 5 2 5   
  • B 𝐴 = 1 4 5  3 βˆ’ 5 2 βˆ’ 8 7 1 0 3 2 7 5 βˆ’ 5 βˆ’ 2 5   
  • CThe matrix has no inverse.
  • D 𝐴 = 1 4 5  3 βˆ’ 8 7 7 5 βˆ’ 5 1 0 βˆ’ 5 2 3 2 βˆ’ 2 5   
  • E 𝐴 = 1 5 6 7  3 βˆ’ 5 2 βˆ’ 8 7 1 0 3 2 7 5 βˆ’ 5 βˆ’ 2 5   

Q4:

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

Find 𝐴.

  • A 𝐴 =  1 4 9 1 6  
  • B 𝐴 =  1 4 9 1 6  
  • C 𝐴 =  7 βˆ’ 6 βˆ’ 9 2 2  
  • D 𝐴 =  βˆ’ 5 βˆ’ 6 9 1 0  
  • E 𝐴 =  7 1 0 1 5 2 2  

Find 𝐴+3𝐴+2𝐼.

  • A 𝐴 + 3 𝐴 + 2 𝐼 =  6 4 6 1 2  
  • B 𝐴 + 3 𝐴 + 2 𝐼 =  0 0 0 0  
  • C 𝐴 + 3 𝐴 + 2 𝐼 =  0 0 0 0  
  • D 𝐴 + 3 𝐴 + 2 𝐼 =  4 1 6 0 6  
  • E 𝐴 + 3 𝐴 + 2 𝐼 =  1 2 0 0 1 2  

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