Fsc Part 1 Mathematics (Complete Solution)

Q2
Find the inverse of the following matrices

(i)   
 
3 -1
2 1
 

Solution:

Let     A=
 
3 -1
1 1
 
Then A-1=
Adj.A
lAl
     lAl=
 
3 -1
1 1
 
=3+2 =5≠0
    Adj.A=
 
1 1
-2 3
 
SO,  A-1=
Adj.A
lAl
=
1
5
 
1 1
-2 3
 
=
 
1/5 1/5
-2/5 3/5
 
Hence the inverse of the matrix =
 
1/5 1/5
-2/5 3/5
 



(ii)   
 
-2  3
-4 5
 

Solution:

Let     A=
 
-2 3
-4 5
 
Then A-1=
Adj.A
lAl
     lAl=
 
-2 3
-4 5
 
=-10+12 =2≠0
    Adj.A=
 
5 -3
4 -2
 
SO,  A-1=
Adj.A
lAl
=
1
2
 
5 -3
4 -2
 
=
 
5/2 -3/2
4/2 -2/2
 
Hence the inverse of the matrix=
 
5/2 -3/2
2 -1
 



(iii)   
 
ι
ι
 

Solution:

Let     A=
 
ι
ι
 
Then A-1=
Adj.A
lAl
     lAl=
 
ι
ι
 
=-2ι22 =-2(-1)-(-1)=2+1=3≠0
    Adj.A=
 
 
SO,  A-1=
Adj.A
lAl
=
1
3
 
 
=
 
-ι/3 -ι/3
-ι/3 2ι/3
 
Hence the inverse of the matrix=
 
-ι/3 -ι/3
-ι/3 2ι/3
 



(iv)   
 
1
6 3
 

Solution:

Let     A=
 
2 1
6 3
 
Then A-1=
Adj.A
lAl
    Adj.A=
 
1 1
-2 3
 
     lAl=
 
2 1
6 3
 
=6-6 =0

Hence , the inverse cannot be determined.

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