Science:Math Exam Resources/Courses/MATH152/April 2022/Question B4 (a)

MATH152 April 2022

  •  QA01  •  QA02  •  QA03  •  QA04  •  QA05  •  QA06  •  QA07  •  QA08  •  QA09  •  QA10  •  QA11  •  QA12  •  QA13  •  QA14  •  QA15  •  QA16  •  QA17  •  QA18  •  QA19  •  QA20  •  QB1 (a)  •  QB1 (b)  •  QB2 (a)  •  QB2 (b)  •  QB2 (c)  •  QB3 (a)  •  QB3 (b)  •  QB4 (a)  •  QB4 (b)  •  QB4 (c)  •  QB5 (a)  •  QB5 (b)  •  QB5 (c)  •  

   Other MATH152 Exams

  •  April 2016  •  April 2015  •  April 2014  •  April 2022  •  April 2011  •  April 2010  •  April 2012  •  April 2013  •  April 2017  •  

Question B4 (a)
Question: Consider the system of differential equations ${\textstyle {\frac {d}{dt}}{\vec {y}}(t)=A{\vec {y}}(t)}$, where $$A=\left[{\begin{array}{cc}3&1\\-2&1\end{array}}\right]$$ (a) Find the eigenvalues and corresponding eigenvectors of ${\textstyle A}$.

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
Recall that the eigenvalues solve the characteristic equation, given by ${\textstyle \det(A-\lambda I)=0}$.

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. To compute the eigenvalues, first form the characteristic equation, then solve for its zeros. $$\left|{\begin{array}{cc}3-\lambda &1\\-2&1-\lambda \end{array}}\right|=(3-\lambda )(1-\lambda )+2=\lambda ^{2}-4\lambda +5=0$$ To solve for the zeros, use $$\lambda ={\frac {-(-4)\pm {\sqrt {(-4)^{2}-4(1)(5)}}}{2(1)}}={\frac {4\pm {\sqrt {-4}}}{2}}=2\pm i,$$ thus the eigenvalues are ${\textstyle \lambda _{1}=2+i}$ and ${\textstyle \lambda _{2}=2-i}$. Since the eigenvalues are complex conjugates of each other, we know that the eigenvectors are also complex conjugates of each other, meaning we only need to compute one eigenvector to know both. In question 17 we found the eigenvectors of a matrix $A$ by solving ${\textstyle A{\vec {v}}=\lambda {\vec {v}}}$ for the components or ${\textstyle {\vec {v}}}$ using basic algebra. For this question we will solve that same system using row operations. Solving for ${\textstyle {\vec {v}}_{1}}$, we write the system as ${\textstyle (A-\lambda _{1}I){\vec {v}}_{1}={\vec {0}}}$, row operations on the augmented system of equations gives $$\left[{\begin{array}{cc|c}1-i&1&0\\-2&-1+i&0\end{array}}\right]\rightarrow \left[{\begin{array}{cc|c}1-i&1&0\\0&-2i&0\end{array}}\right]\rightarrow \left[{\begin{array}{cc|c}1-i&1&0\\0&0&0\end{array}}\right]$$ If ${\textstyle {\vec {v}}_{1}=[v_{1}\ \ v_{2}]^{T}}$, then we have ${\textstyle (1-i)v_{1}+v_{2}=0}$. Choosing (arbitrarily) ${\textstyle v_{1}=1}$, we get ${\textstyle v_{2}=-1+i}$, and the two eigenvectors are $${\vec {v}}_{1}=\left[{\begin{array}{c}1\\-1+i\end{array}}\right],\ \ {\vec {v}}_{2}=\left[{\begin{array}{c}1\\-1-i\end{array}}\right].$$