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

From Grad Wiki
Jump to navigation Jump to search
(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 \\...")
 
Line 1: Line 1:
−
<span class="exam">Consider the matrix &nbsp;<math style="vertical-align: -31px">A=     
+
<span class="exam"> Find the eigenvalues and eigenvectors of the matrix &nbsp;<math style="vertical-align: -31px">A=     
 
     \begin{bmatrix}
 
     \begin{bmatrix}
−
           1 & -4 & 9 & -7 \\
+
           1 & 1 & 1 \\
−
          -1 & 2  & -4 & 1 \\
+
           0 & -1  & 1 \\
−
           5 & -6 & 10 & 7
+
           0 & 0 & 2
−
        \end{bmatrix}</math>&nbsp; and assume that it is row equivalent to the matrix
+
         \end{bmatrix}.</math>
−
 
 
−
::<math>B=   
 
−
    \begin{bmatrix}
 
−
          1 & 0 & -1 & 5 \\
 
−
          0 & -2 & 5 & -6 \\
 
−
           0 & 0 & 0 & 0
 
−
         \end{bmatrix}.</math>     
 
−
   
 
−
<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>
 
−
 
 
−
<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>
 
  
  
Line 26: Line 15:
  
 
'''Solution:'''
 
'''Solution:'''
−
 
−
'''(a)'''
 
−
 
−
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
−
!Step 1: &nbsp;
 
−
|-
 
−
|
 
−
|}
 
−
 
−
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
−
!Step 2: &nbsp;
 
−
|-
 
−
|
 
−
|}
 
−
 
−
'''(b)'''
 
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
Line 59: Line 32:
 
!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
|-
 
|-
−
|&nbsp;&nbsp; '''(a)''' &nbsp; &nbsp;
+
|&nbsp;&nbsp; &nbsp; &nbsp;  
−
|-
 
−
|&nbsp;&nbsp; '''(b)''' &nbsp; &nbsp;  
 
 
|}
 
|}
 
[[031_Review_Part_3|'''<u>Return to Sample Exam</u>''']]
 
[[031_Review_Part_3|'''<u>Return to Sample Exam</u>''']]

Revision as of 19:22, 9 October 2017

Find the eigenvalues and eigenvectors of the matrix  


Foundations:  


Solution:

Step 1:  
Step 2:  


Final Answer:  
      

Return to Sample Exam

Retrieved from "https://gradwiki.math.ucr.edu/index.php?title=031_Review_Part_3,_Problem_2&oldid=3570"