Fsc Part 1 Mathematics (Complete Solution)

Q2
Show that

If  A=
 
1 2 0
3 2 -1
-1 3 2
 
 , show that 

(i) A+At is symmetric

SOLUTION:

  A+At =  
 
1 2 0
3 2 -1
-1 3 2
 
  + 
 
3 -1
2 2 3
0 -1 2
 
 
       =  
 
2 5 -1
5 4 2
-1 2 4
 
      (A+At )t =  
 
2 5 -1
5 4 2
-1 2 4
 

=A+At

Hence (A+At ) is symmetric.



(ii) A-At is skew-symmetric.

SOLUTION:

  A-At =  
 
1 2 0
3 2 -1
-1 3 2
 
  - 
 
3 -1
2 2 3
0 -1 2
 
 
       =  
 
0 -1 1
1 0 -4
-1 4 0
 
      (A-At )t =  
 
0 1 -1
-1 0 4
1 -4 0
 
       =-  
 
0 -1 1
1 0 -4
-1 4 0
 

=-(A-At)

Hence A-At  is skew symmetric.

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