Fsc Part 1 Mathematics (Complete Solution)

Q12
Find the matrix x if;

(i)    X
 
5 2
-2 1
 
 =  
 
-1 5
12 3
 
 

Solution:

    X
 
5 2
-2 1
 
 =  
 
-1 5
12 3
 
 
Let  X=
 
a b
c d
 
  then 
   
 
b
c d
 
  
 
5 2
-2 1
 
   =  
 
-1 5
12 3
 
   
 
5a-2b  2a+b
5c-2d 2c+d
 
 = 
 
-1 5
12 3
 

Comparing the corresponding elements, we get

5a-2b=-1   ...............(i)

2a+b=5     ...............(ii)

5c-2d=12   ................(iii)

2c+d=3      ................(iv)

to solve (i) and (ii), put b=5-2a from (ii) in(i) then

5a-2(5-2a)=-1

5a-10+4a=-1

9a=9

=> a=1

put a=-1 in(ii) , then

b=5-2a

b=5-2(1)

b=5-2

b=3

to solve (iii) and (iv), put d=3-2c from (iv) in (iii), then

5c-2(3-2c)=12

5c-6+4c=12

9c=18

c=2

put c=2 in (iv) , then

d=3-2c=5-2(2)=3-4=-1

d=-1

 Thus    X=
 
b
c d
 
 =  
 
1 3
2 -1
 

 

(ii)    
 
5 2
-2 1
 
 X=  
 
2 1
5 10
 
 

Solution:

    
 
5 2
-2 1
 
X =  
 
2 1
5 10
 
 
Let  X=
 
a b
c d
 
  then 
   
 
5 2
-2 1
 
  
 
a b
c d
 
   =  
 
2 1
5 10
 
   
 
5a-2c  5b+2d
-2a+c -2b+d
 
 = 
 
2 1
5 10
 

Comparing the corresponding elements, we get

5a+2c=2  ...............(i)

5b+2d=1    ...............(ii)

-2a+c=5   ................(iii)

-2b+d=10      ................(iv)

to solve (i) and (iii), put c=5+2a from (iii) in (i) , then

5a+2(5+2a)=2

=> 5a+10+4a=2

9a=-8

a=-8/9

put a=-8/9   in (iii) , then

c=5+2a

c=5+2(-8/9)

c=29/9

to solve (ii) and (iv) , put d=10+2b from (iv) in (ii), then

5b+2(10+2b)=1

=> 5b+20+4b=1

9b=-19

b=-19/9

put b=-19/9  in (ii), then

d=10+2b

d=10+2(-19/9)

d=10-38/9

d=52/9

 Thus    X=
 
b
c d
 
 =  
 
-8/9 -19/9
29/9 52/9
 
 

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