Fsc Part 1 Mathematics (Complete Solution)

Q14
Verify that

Verify that (AB)-1=B-1A-1 if



(i)  A =
 
1 2
-1 0
 
 ,  B=
 
-3 1
4 -1
 
 

SOLUTION: Now taking L.H.S

 AB=
 
1 2
-1 0
 
  
 
-3 1
4 -1
 
   =
 
-3+8 1-2
3+0 -1+0
 
    
   =
 
5 -1
3 -1
 
    
lABl= 
 
5 -1
3 -1
 

=-5+3

=-2

  (AB)-1=
Adj.AB
lABl
=
1
-2
 
-1 1
-3 5
 
=
1
2
 
1 -1
3 -5
 
 
   =
 
1/2 -1/2
3/2 -5/2
 
   .........(i)    

Taking R.H.S

lAl= 
 
1 2
-1 -0
 

=0+2

=2

  A-1=
Adj.A
lAl
=
1
2
 
0 -2
1 1
 
 
0 -1
1/2 1/2
 
lBl= 
 
-1 1
4 -3
 

=3-4

=-1

  B-1=
Adj.B
lBl
=
1
-1
 
-1 -1
-4 -3
 
 
1 1
4 3
 
 B-1A-1=
 
1
4 3
 
  
 
0 -1
1/2 1/2
 
   =
 
(0+1)/2 (-2+1)/2
(0+3)/2 (-8+3)/2
 
    
   =
 
1/2 -1/2
3/2 -5/2
 
   .........(ii)   

(i)=(ii)

L.H.S=R.H.S

Hence (AB)-1=B-1A-1

SO PROVED



(ii)  A =
 
1
2 2
 
 ,  B=
 
4 3
2 1
 
 

SOLUTION: Now taking L.H.S

 AB=
 
5 1
2 2
 
  
 
4 3
2 1
 
   =
 
20+2 15+1
8+4 6+2
 
    
   =
 
22 16
10 8
 
    
lABl= 
 
22 16
12 8
 

= 176-192

= -16

  (AB)-1=
Adj.AB
lABl
=
1
-16
 
8 -16
-12 22
 
=
1
16
 
-8 16
12 -22
 
 
   =
 
-1/2 1
3/4 -11/8
 
   .........(i)    

Taking R.H.S

lAl= 
 
5 1
2 2
 

=10-2

=8

  A-1=
Adj.A
lAl
=
1
8
 
2 -1
-2 5
 
 
1/4 -1/8
-1/4 5/8
 
lBl= 
 
4 3
2 1
 

=4-6

=-2

  B-1=
Adj.B
lBl
=
1
-2
 
1 -3
-2 4
 
 
-1/2 3/2
1 -2
 
 B-1A-1=
 
-1/2 3/2
1 -2
 
  
 
1/4 -1/8
-1/4 5/8
 
   =
 
(-2-6)/16 (1+15)/16
(4+8)/16 (-2-20)/16
 
    
   =
 
-1/2 1
3/4 -11/8
 
   .........(ii)   

(i)=(ii)

L.H.S=R.H.S

Hence (AB)-1=B-1A-1

SO PROVED

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