Difference between revisions of "031 Review Part 3, Problem 8"

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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">Give an example of a &nbsp;<math style="vertical-align: 0px">3\times 3</math>&nbsp; matrix &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; with eigenvalues 5,-1 and 3.
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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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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Foundations: &nbsp;  
 
!Foundations: &nbsp;  
 
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|The eigenvalues of a diagonal matrix are the entries on the diagonal.
 
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'''Solution:'''
 
'''Solution:'''
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'''(a)'''
 
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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!Step 1: &nbsp;  
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! &nbsp;  
 
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|One example of such a matrix is
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|}
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
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!Step 2: &nbsp;
 
 
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::<math>A=\left[\begin{array}{ccc}  
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          5 & 0 & 0\\
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'''(b)'''
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            0 & -1 & 0\\
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            0 & 0 & 3
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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        \end{array}\right].</math>
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!Step 1: &nbsp;
 
 
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|Since &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is a diagonal matrix, the eigenvalues of &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; are the entries on the diagonal.
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|}
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
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!Step 2: &nbsp;
 
 
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|Hence, the eigenvalues of &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; are &nbsp;<math style="vertical-align: -4px">5,-1,3.</math>
 
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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; &nbsp; &nbsp; One example is &nbsp;<math style="vertical-align: -31px">A=\left[\begin{array}{ccc} 
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          5 & 0 & 0\\
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|&nbsp;&nbsp; '''(b)''' &nbsp; &nbsp;
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            0 & -1 & 0\\
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            0 & 0 & 3
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        \end{array}\right].</math>
 
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|}
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[[031_Review_Part_3|'''<u>Return to Sample Exam</u>''']]
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[[031_Review_Part_3|'''<u>Return to Review Problems</u>''']]

Latest revision as of 14:06, 15 October 2017

Give an example of a    matrix    with eigenvalues 5,-1 and 3.

Foundations:  
The eigenvalues of a diagonal matrix are the entries on the diagonal.


Solution:

 
One example of such a matrix is
Since    is a diagonal matrix, the eigenvalues of    are the entries on the diagonal.
Hence, the eigenvalues of    are  


Final Answer:  
       One example is  

Return to Review Problems

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