Science:Math Exam Resources/Courses/MATH221/December 2008/Question 07 (b)

MATH221 December 2008

Work in progress: this question page is incomplete, there might be mistakes in the material you are seeing here.

  •  Q1 (a)  •  Q1 (b)  •  Q2 (a)  •  Q2 (b)  •  Q3 (a)  •  Q3 (b)  •  Q4 (a)  •  Q4 (b)  •  Q5 (a)  •  Q5 (b)  •  Q6 (a)  •  Q6 (b)  •  Q7 (a)  •  Q7 (b)  •  Q8 (a)  •  Q8 (b)  •  Q9 (a)  •  Q9 (b)  •  Q10 (a)  •  Q10 (b)  •  

   Other MATH221 Exams

  •  December 2008  •  December 2009  •  December 2007  •  April 2010  •  April 2009  •  December 2011  •  April 2013  •  

Question 07 (b)
True or false (explain your answer): If A and B are similar matrices, then A and B have the same eigenvalues.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
If two matrices are similar, then there is a relation between their respective characteristic polynomials (what is it?). How does the characteristic polynomial of a matrix relate to its eigenvalues?

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. The statement is true: If A and B are similar, then there is an invertible matrix P for which $A=P^{-1}BP$. Then we can write ${\begin{aligned}xI-A&=xI-P^{-1}BP\\&=P^{-1}(xI-B)P.\end{aligned}}$ Therefore we compute that $\det(xI-A)=\det(P^{-1}(xI-B)P)=\det(P)^{-1}\det(xI-B)\det(P)=\det(xI-B)$. This equality means precisely that the characteristic polynomials for A and B are identical. Since the eigenvalues of a matrix are the zeros of its characteristic polynomial, and the characteristic polynomials for A and B are identical, we conclude that the eigenvalues of A and B are the same.

Click here for similar questions

MER RH flag, MER RQ flag, MER RS flag, MER RT flag, MER Tag Eigenvalues and eigenvectors, MER Tag Matrix similarity, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag