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Worksheet: Eigenvectors of a Matrix

Q1:

Suppose one eigenvalue of a real matrix 𝐴 is πœ† = βˆ’ 1 + 4 𝑖 1 and the corresponding eigenvector is Which of the following must also be an eigenpair of 𝐴 ?

  • A πœ† = βˆ’ πœ† 2 1 and 𝑣 = βˆ’ 𝑣 2 1
  • B πœ† = βˆ’ 1 + 4 𝑖 2 and 𝑣 =  3 + 2 𝑖 βˆ’ 1  2
  • C πœ† = βˆ’ 1 βˆ’ 4 𝑖 2 and 𝑣 =  βˆ’ 3 βˆ’ 2 𝑖 βˆ’ 1  2
  • D πœ† = βˆ’ 1 βˆ’ 4 𝑖 2 and 𝑣 =  3 βˆ’ 2 𝑖 βˆ’ 1  2

Q2:

Suppose that 𝑣 is an eigenvector of a matrix 𝐴 corresponding to a nonzero eigenvalue πœ† . Is 𝐴 π‘₯ = 𝑣 always solvable?

  • AYes
  • BNo