text: # Eigenvalue true and false
  
  1. Dec 6, 2009
  
  ### shiri
  
  Let A and B be nxn matrices, where B is invertible. Suppose that 4 is an eigenvalue of A, and 5 is an eigenvalue of B. Find ALL true statements.
  
  A) 4 is an eigenvalue of A^T
  B) 4 is an eigenvalue of (B^−1)AB
  C) 265 is an eigenvalue of (A^4)+A+5I
  D) 8 is an eigenvalue of A+(A^T)
  E) 20 is an eigenvalue of AB
  F) None of the above
  
  I choose:
  A)
  B)
  
  am I right?
  
  Last edited: Dec 6, 2009
  2. Dec 7, 2009
  
  ### omega101
  
  Re: Eigenvalues
  
  Let A and B be nn matrices, where B is invertible. Suppose that 5 is an eigenvalue of A, and 4 is an eigenvalue of B. Find ALL true statements below.
  A. 20 is an eigenvalue of AB
  B. 10 is an eigenvalue of A+AT
  C. 34 is an eigenvalue of A2+A+4I
  D. 5 is an eigenvalue of AT
  E. 5 is an eigenvalue of B−1AB
  F. None of the above
  
  HELP!!!
  
  3. Dec 8, 2009
  
  ### shiri
  
  Re: Eigenvalues
  
  anyone?
  
  4. Dec 8, 2009
  
  ### Dick
  
  Re: Eigenvalues
  
  If you think a statement is true you should probably give a reason. If you think it's not true then you should figure out a counterexample. Otherwise you are just playing a guessing game.
  
  5. Dec 8, 2009
  
  ### shiri
  
  Re: Eigenvalues
  
  A) Say it was a 3x3 matrix. If I transpose the matrix, the eigenvalue still be 4.
  B) BB^-1 becomes an identity and it lefts with A only.
  C) A^4 + A + 5I = (4)^4 + 4 + 5 = 265. However, I'm not too sure about this statement.
  D) Not too sure if I can simply add those two together. If I assume that upper and lower triangular part are zeros.
  E) Obviously not a correct statement.
  
  What do you think, Dick? A & B only?
  
  I am very skeptical about C & D. Please help me on this, Dick.
  
  6. Dec 8, 2009
  
  ### Dick
  
  Re: Eigenvalues
  
  Ok, for A. A better reason is that the eigenvalues are the roots of det(A-xI) and determinant of the transpose of A-xI is the same as the determinant of A-xI. Your reason for B isn't so good. Matrices in general don't commute. If v is the eigenvector of A then Av=4v. Define a vector y=B^(-1)x. What's (B^(-1)AB)y? For C, use Av=4v and figure out what (A^4+A+5I)v is. What does that tell you about the eigenvalues of (A^4+A+5I)? Can you think of a counterexample for D and E?
  
  7. Dec 8, 2009
  
  ### shiri
  
  Re: Eigenvalues
  
  A, B, C are true
  
  D) If I assume there are non-zeros in the matrix
  
  eg.
  |1 4|+|1 3|=|2 7|
  |3 2| |4 2| |7 4|
  
  So, eval are roots of different roots and I have to use a quadratic formula to find those roots
  
  E) Multiplying A&B will not come out as 20 b/c the matrix should be multiplied, not multiplying scalars (eigenvalues). So, False.
  
  What do you think?
  
  8. Dec 8, 2009
  
  ### Dick
  
  Re: Eigenvalues
  
  Yes, A, B and C are true. D and E are only true for some matrices, not for others. I'll give you examples where they aren't and you figure out why, ok? I can't really tell what you are trying to say for either of them. For D take A=[[4,1],[0,0]]. What are the eigenvalues of A, A^(T) and A+A^(T)? For E take A=[[4,0],[0,0]] and B=[[0,0],[0,5]]. What are the eigenvalues of A, B and AB? If you want another exercise, figure out specific matrices where D and E ARE true.
explanation: The document contains a math question and an answer to this question. The question is about determining which statements are true regarding eigenvalues of matrices A and B, given that 4 is an eigenvalue of A and 5 is an eigenvalue of B. The user provides their reasoning and selections for the statements, and another user (Dick) provides feedback and further explanations.
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  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
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distilabel_metadata.role: system
distilabel_metadata.content: # Eigenvalue true and false
  
  1. Dec 6, 2009
  
  ### shiri
  
  Let A and B be nxn matrices, where B is invertible. Suppose that 4 is an eigenvalue of A, and 5 is an eigenvalue of B. Find ALL true statements.
  
  A) 4 is an eigenvalue of A^T
  B) 4 is an eigenvalue of (B^−1)AB
  C) 265 is an eigenvalue of (A^4)+A+5I
  D) 8 is an eigenvalue of A+(A^T)
  E) 20 is an eigenvalue of AB
  F) None of the above
  
  I choose:
  A)
  B)
  
  am I right?
  
  Last edited: Dec 6, 2009
  2. Dec 7, 2009
  
  ### omega101
  
  Re: Eigenvalues
  
  Let A and B be nn matrices, where B is invertible. Suppose that 5 is an eigenvalue of A, and 4 is an eigenvalue of B. Find ALL true statements below.
  A. 20 is an eigenvalue of AB
  B. 10 is an eigenvalue of A+AT
  C. 34 is an eigenvalue of A2+A+4I
  D. 5 is an eigenvalue of AT
  E. 5 is an eigenvalue of B−1AB
  F. None of the above
  
  HELP!!!
  
  3. Dec 8, 2009
  
  ### shiri
  
  Re: Eigenvalues
  
  anyone?
  
  4. Dec 8, 2009
  
  ### Dick
  
  Re: Eigenvalues
  
  If you think a statement is true you should probably give a reason. If you think it's not true then you should figure out a counterexample. Otherwise you are just playing a guessing game.
  
  5. Dec 8, 2009
  
  ### shiri
  
  Re: Eigenvalues
  
  A) Say it was a 3x3 matrix. If I transpose the matrix, the eigenvalue still be 4.
  B) BB^-1 becomes an identity and it lefts with A only.
  C) A^4 + A + 5I = (4)^4 + 4 + 5 = 265. However, I'm not too sure about this statement.
  D) Not too sure if I can simply add those two together. If I assume that upper and lower triangular part are zeros.
  E) Obviously not a correct statement.
  
  What do you think, Dick? A & B only?
  
  I am very skeptical about C & D. Please help me on this, Dick.
  
  6. Dec 8, 2009
  
  ### Dick
  
  Re: Eigenvalues
  
  Ok, for A. A better reason is that the eigenvalues are the roots of det(A-xI) and determinant of the transpose of A-xI is the same as the determinant of A-xI. Your reason for B isn't so good. Matrices in general don't commute. If v is the eigenvector of A then Av=4v. Define a vector y=B^(-1)x. What's (B^(-1)AB)y? For C, use Av=4v and figure out what (A^4+A+5I)v is. What does that tell you about the eigenvalues of (A^4+A+5I)? Can you think of a counterexample for D and E?
  
  7. Dec 8, 2009
  
  ### shiri
  
  Re: Eigenvalues
  
  A, B, C are true
  
  D) If I assume there are non-zeros in the matrix
  
  eg.
  |1 4|+|1 3|=|2 7|
  |3 2| |4 2| |7 4|
  
  So, eval are roots of different roots and I have to use a quadratic formula to find those roots
  
  E) Multiplying A&B will not come out as 20 b/c the matrix should be multiplied, not multiplying scalars (eigenvalues). So, False.
  
  What do you think?
  
  8. Dec 8, 2009
  
  ### Dick
  
  Re: Eigenvalues
  
  Yes, A, B and C are true. D and E are only true for some matrices, not for others. I'll give you examples where they aren't and you figure out why, ok? I can't really tell what you are trying to say for either of them. For D take A=[[4,1],[0,0]]. What are the eigenvalues of A, A^(T) and A+A^(T)? For E take A=[[4,0],[0,0]] and B=[[0,0],[0,5]]. What are the eigenvalues of A, B and AB? If you want another exercise, figure out specific matrices where D and E ARE true.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and an answer to this question. The question is about determining which statements are true regarding eigenvalues of matrices A and B, given that 4 is an eigenvalue of A and 5 is an eigenvalue of B. The user provides their reasoning and selections for the statements, and another user (Dick) provides feedback and further explanations.
  
  Label: question-and-answer