Difference between revisions of "031 Review Part 3, Problem 2"

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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"> Find the eigenvalues and eigenvectors of the matrix &nbsp;<math style="vertical-align: -31px">A=     
 
     \begin{bmatrix}
 
     \begin{bmatrix}
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           1 & -4 & 9 & -7 \\
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           1 & 1 & 1 \\
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          -1 & 2  & -4 & 1 \\
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           0 & -1  & 1 \\
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           5 & -6 & 10 & 7
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           0 & 0 & 2
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        \end{bmatrix}</math>&nbsp; and assume that it is row equivalent to the matrix
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         \end{bmatrix}.</math>
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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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|-
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|
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|An eigenvector of a matrix &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is a nonzero vector &nbsp;<math style="vertical-align: 0px">\vec{x}</math>&nbsp; such that &nbsp;<math style="vertical-align: 0px">A\vec{x}=\lambda\vec{x}</math>&nbsp; for some scalar &nbsp;<math style="vertical-align: 0px">\lambda.</math>
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|-
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|In this case, we say that &nbsp;<math style="vertical-align: 0px">\lambda</math>&nbsp; is an eigenvalue of &nbsp;<math style="vertical-align: 0px">A.</math>
 
|}
 
|}
  
  
 
'''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;"
 
!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
|-
 
|-
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|
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|Since &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is a triangular 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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|So, the eigenvalues of &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; are &nbsp;<math style="vertical-align: -4px">1,-1,</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">2.</math>
 
|}
 
|}
  
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
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|-
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|
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|Since the matrix is triangular and all the eigenvalues are distinct, the eigenvectors of &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; are
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|}
 
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'''(b)'''
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
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!Step 1: &nbsp;  
 
 
|-
 
|-
 
|
 
|
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|}
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::<math>\begin{bmatrix}
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          1  \\
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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          0 \\
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!Step 2: &nbsp;
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          0
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        \end{bmatrix},\begin{bmatrix}
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          0  \\
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          1 \\
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          0
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        \end{bmatrix},\begin{bmatrix}
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          0  \\
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          0 \\
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          1
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        \end{bmatrix}</math>
 
|-
 
|-
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|
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|where each eigenvector has eigenvalue &nbsp;<math style="vertical-align: -4px">1,-1</math>&nbsp; and &nbsp;<math style="vertical-align: -4px">2,</math>&nbsp; respectively.
 
|}
 
|}
  
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
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|-
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|&nbsp;&nbsp; '''(a)''' &nbsp; &nbsp;  
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|&nbsp;&nbsp; &nbsp; &nbsp; The eigenvalues of &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; are &nbsp;<math style="vertical-align: -4px">1,-1</math>&nbsp; and &nbsp;<math style="vertical-align: -4px">2,</math>&nbsp; and the corresponding eigenvectors are
 
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|-
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|&nbsp;&nbsp; '''(b)''' &nbsp; &nbsp;
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|
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::<math>\begin{bmatrix}
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          1  \\
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          0 \\
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          0
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        \end{bmatrix},\begin{bmatrix}
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          0  \\
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          1 \\
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          0
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        \end{bmatrix},\begin{bmatrix}
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          0  \\
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          0 \\
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          1
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        \end{bmatrix}.</math>
 
|}
 
|}
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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 13:53, 15 October 2017

Find the eigenvalues and eigenvectors of the matrix  

Foundations:  
An eigenvector of a matrix    is a nonzero vector  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\vec {x}}}   such that  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A{\vec {x}}=\lambda {\vec {x}}}   for some scalar  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lambda .}
In this case, we say that    is an eigenvalue of  


Solution:

Step 1:  
Since    is a triangular matrix, the eigenvalues of    are the entries on the diagonal.
So, the eigenvalues of    are  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1,-1,}   and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2.}
Step 2:  
Since the matrix is triangular and all the eigenvalues are distinct, the eigenvectors of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}   are
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix},\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix},\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}}
where each eigenvector has eigenvalue  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1,-1}   and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2,}   respectively.


Final Answer:  
       The eigenvalues of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}   are  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1,-1}   and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2,}   and the corresponding eigenvectors are
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix},\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix},\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}.}

Return to Review Problems

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