Fsc Part 1 Mathematics (Complete Solution)

Q13
Find the matrix a if ,

(i)    
 
5 -1
0 0
3 1
 
 A=  
 
3 -7
0 0
7 2
 
 

Solution:

    
 
5 2
-2 1
 
A =  
 
2 1
5 10
 
 
Let  A=
 
a b
c d
 
  then 
   
 
5 -1 
0 0
3 1
 
  
 
a b
c d
 
   =  
 
3 -7
0 0
7 2
 
   
 
5a-c  5b-d
0+0 0+0
3a+c 3b+d
 
 = 
 
3 -7
0 0
7 2
 

Comparing the corresponding elements, we get

5a-c=3 ......(i)

5b-d=-7 ......(ii)

3a+c=7    .....(iii)

3b+d=2  ........(iv)

adding (i) and (iii), we get

8a=10

=> a=5/4

(iii) gives c=7-3a

=7-15/4

c=13/4

adding (ii) and (iv) , we get

8b=-5

b=-5/8

(iv) gives d=2-3b

d=2+15/8

d=31/8

 Thus    A=
 
b
c d
 
 =  
 
5/4 -5/8
13/4 31/8
 
 


(ii)    
 
-1
-1 2
 
 A=  
 
0 -3 8
3 3 -7
 
 

Solution:

    
 
2 -1
-1 2
 
A =  
 
0 -3 8
3 3 -7 
 
 
Let  A=
 
a b c
e f
 
  then 
   
 
2 -1 
-1 2
 
  
 
a b c
e f
 
   =  
 
0 -3 8
3 3 -7
 
   
 
2a-d  2b-e 2c-f
-a+2d -b+2e -c+2f
 
 = 
 
-3 8
3 3 -7
 

Comparing the corresponding elements, we get

2a-d=0 .........(i)

2b-e=-3  ..........(ii)

2c-f=8 .........(iii)

-a+2d=3 ...............(iv)

-b+2e=3   ................(v)

-c+2f=-7 ........(vi)

To solve (i) and (iv) , put d=2a from (i) in (iv) , then

-a+2d=3

-a+2(2a)=3

-a+4a=3

3a=3

a=1

put a=1 in (iv), then

d=2a

d=2(1)

d=2

to solve (ii) and (v) , put e=2b+3 from (ii) in (v), then

-b+2(2b+3)=3

-b+4b+6=3

3b=-3

b=-1

put b=-1 in (v) , then

e=2b+3

e=2(-1)+3

e=-2+3

e=1

to solve (iii) and (vi), put f=2c-8 from (iii) in (vi), then

-c+2f=-7

-c+2(2c-8)=-7

3c=9

c=3

put c=3 in (vi) , then

f=2c-8

f=2(3)-8

f=6-8

f=-2

 Thus    A=
 
b c
e f
 
 =  
 
1 -1 3
2 1 -2
 
 

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