Fsc Part 1 Mathematics (Complete Solution)

Q6
Show That

If  A=
 
ι 1+ι
1
 
 , show that



(i) A + ( A ) is hermitian

SOLUTION:

A =  
 
1-ι
1 ι
 
(A)t =  
 
1
1-ι ι
 
A + (A)t =  
 
ι 1+ι
1
 
 + 
 
1
1-ι ι
 
 
       =   
 
0 2+ι
2-ι 0
 
A + (A)t =
 
0 2 - ι
2+ι 0
 
(A + (A)t)t =
 
0 2+ι
2-ι 0
 


= A + ( A )t


= Thus A + ( A )t is hermitian.




(ii) A - ( A )t is hermitian

SOLUTION:


A =   
 
1-ι
1 ι
 
(A)t=  
 
1
1-ι ι
 
A-(A)t=  
 
ι 1+ι
1
 
  - 
 
1
1-ι ι
 
 
       =  
 
ι
ι -2ι
 
A-(A)t=
 
-2ι
 
(A-(A)t)t=
 
-2ι
 
= -
 
ι
ι -2ι
 


= -(A - ( A )t)


Thus A - ( A )t is skew hermitian.

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