Science:Math Exam Resources/Courses/MATH221/April 2009/Question 12 (b)

MATH221 April 2009

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

  •  Q1  •  Q2  •  Q3  •  Q4  •  Q5  •  Q6  •  Q7 (a)  •  Q7 (b)  •  Q8  •  Q9 (a)  •  Q9 (b)  •  Q9 (c)  •  Q9 (d)  •  Q10  •  Q11 (a)  •  Q11 (b)  •  Q11 (c)  •  Q12 (a)  •  Q12 (b)  •  Q12 (c)  •  Q12 (d)  •  Q12 (e)  •  Q12 (f)  •  Q12 (g)  •  Q12 (h)  •  Q12 (i)  •  Q12 (j)  •  

   Other MATH221 Exams

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

Question 12 (b)
Every invertible matrix can be diagonalized.

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

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. This is false. To construct a counterexample, we need to think of an invertible matrix which cannot be diagonalized, i.e. an invertible matrix which has only one linearly independent eigenvector. The first invertible matrix which comes to mind is the identity matrix, ${\begin{pmatrix}1&0\\0&1\end{pmatrix}},$ however, this is not a counterexample as it is diagonal. However, consider the simple modification $A={\begin{pmatrix}1&1\\0&1\end{pmatrix}}.$ A is invertible as its determinant is 1, however, A is not diagonalizable as the only linearly independent solution to the eigenvector equation ${\begin{pmatrix}0&1\\0&0\end{pmatrix}}{\begin{pmatrix}v_{1}\\v_{2}\end{pmatrix}}={\begin{pmatrix}0\\0\end{pmatrix}}$ is ${\begin{pmatrix}v_{1}\\v_{2}\end{pmatrix}}={\begin{pmatrix}1\\0\end{pmatrix}}$, and so A has only one linearly independent eigenvector corresponding to its eigenvalue 1. Therefore, we conclude ${\begin{pmatrix}1&1\\0&1\end{pmatrix}}$ is a counterexample to the statement.

Click here for similar questions

MER CH flag, MER QGQ flag, MER RS flag, MER RT flag, MER Tag Invertibility, MER Tag Matrix diagonalization, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag