Please do question #2 and #3Show all work, that if first university student will understand it, do...

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Please do question #2 and #3Show all work, that if first university student will understand it, do all steps , there is a sample of (-18 Q2. X (a) Given that one eigenvector of the matrix B := 15 5 — XXXXXXXXXX —174 —55 is ( 2 2 ) , 1
find one eigenvalue and then the other two eigenvalues. [5] (b) Find an eigenvector of B that does not belong to a dominant eigenvalue by using row operations. [5] (c) Explain several sorts of different behaviour we could expect for Bkvo as k increases with respect to these particular eigenvalues and different choices for V. [2]
Q3. (a) Which vector space axioms are false or true for these two different sets of points in the (x, y) plane? (give reasons or counterexamples for each axiom/set) [5] (i) y < 2x + 1 (ii) y > ix]
(5 y o — (b) Create two lines which are wholly within in the plane P := x2 = 11 z 3 but which do not ever intersect. [2] (c) Find the equation of a line L within P which is perpendicular to your lines from (b) and determine where L intersects with them both, and hence determine the distance between the lines. [5]
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David · Accepted Answer

Sol: 
(a) Given matrix B and one eigenvector 
  (
          
        
      
+         (
 
 
 
+ 
    (
          
        
      
+(
 
 
 
+  (
 
 
 
+   (
 
 
 
+       
Hence the eigenvalue corresponding to   is 3 
The characteristic equation of the matrix B is given by, 
|    |    
|
            
          
        
|
(   )(      )    
(   )(   )(   )    
The eigenvalues are 3, -3, and 2. 
(b) For       
(
          
        
      
+(
   
   
   
+    
Using row operations on the matrix,  
    
 
 
(      ) 
(
          
        
    
+ 
    
 
 
(     )  
(
          
    
    
+
(
          
    
   
+ 
    
 
  
(        ) 
(
    
    
   
+
Hence    is given by,
(
 

 

 
 
 )
 

For      
(
          
        
      
+(
   
   
   
+    
Using row operations,

Please do question #2 and #3Show all work, that if first university student will understand it, do all steps , there is a sample of (-18 Q2. X (a) Given that one eigenvector of the matrix B := 15 5 —...

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