Difference between revisions of "031 Review Part 3, Problem 10"

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(Created page with "<span class="exam">Consider the matrix  <math style="vertical-align: -31px">A= \begin{bmatrix} 1 & -4 & 9 & -7 \\ -1 & 2 & -4 & 1 \\...")
 
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<span class="exam">Consider the matrix &nbsp;<math style="vertical-align: -31px">A=   
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<span class="exam">Show that if &nbsp;<math style="vertical-align: 0px">\vec{x}</math>&nbsp; is an eigenvector of the matrix product &nbsp;<math style="vertical-align: 0px">AB</math>&nbsp; and &nbsp;<math style="vertical-align: -5px">B\vec{x}\ne \vec{0},</math>&nbsp; then &nbsp;<math style="vertical-align: 0px">B\vec{x}</math>&nbsp; is an eigenvector of &nbsp;<math style="vertical-align: 0px">BA.</math>
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    \begin{bmatrix}
 
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          1 & -4 & 9 & -7 \\
 
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          -1 & 2  & -4 & 1 \\
 
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          5 & -6 & 10 & 7
 
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        \end{bmatrix}</math>&nbsp; and assume that it is row equivalent to the matrix  
 
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::<math>B=   
 
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    \begin{bmatrix}
 
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          1 & 0 & -1 & 5 \\
 
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          0 & -2  & 5 & -6 \\
 
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          0 & 0 & 0 & 0
 
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        \end{bmatrix}.</math>     
 
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<span class="exam">(a) List rank &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">\text{dim Nul }A.</math>
 
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<span class="exam">(b) Find bases for &nbsp;<math style="vertical-align: 0px">\text{Col }A</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">\text{Nul }A.</math>&nbsp; Find an example of a nonzero vector that belongs to &nbsp;<math style="vertical-align: -5px">\text{Col }A,</math>&nbsp; as well as an example of a nonzero vector that belongs to &nbsp;<math style="vertical-align: 0px">\text{Nul }A.</math>
 
  
  
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'''Solution:'''
 
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'''(a)'''
 
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!Step 1: &nbsp;
 
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!Step 2: &nbsp;
 
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'''(b)'''
 
  
 
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
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|&nbsp;&nbsp; '''(a)''' &nbsp; &nbsp;
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|&nbsp;&nbsp; '''(b)''' &nbsp; &nbsp;  
 
 
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[[031_Review_Part_3|'''<u>Return to Sample Exam</u>''']]
 
[[031_Review_Part_3|'''<u>Return to Sample Exam</u>''']]

Revision as of 19:31, 9 October 2017

Show that if    is an eigenvector of the matrix product    and    then    is an eigenvector of  


Foundations:  


Solution:

Step 1:  
Step 2:  


Final Answer:  
      

Return to Sample Exam

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