Fsc Part 1 Mathematics (Complete Solution)

Q12
Which of the following are matrices are singular and which of them are non singular?

 i) 
 
1 0 3
3 1 -1
0 2 4
 

SOLUTION:

Let   A= 
 
1 0 3
3 1 -1
0 2 4
 
lAl = 
 
1 0 3
3 1 -1
0 2 4
 
 =  1
 
1 -1
2 4
 
  - 0 
 
3 -1
0 4
 
  +  3 
 
3 1
0 2
 

=1(4+2)-0+3(6-0)

=6+18

=4

≠0

So A is an singular

 ii) 
 
2 3 -1
1 1 0
2 -3 5
 

SOLUTION:

Let   A= 
 
2 3 -1
1 1 0
2 -3 5
 
lAl = 
 
2 3 -1
1 1 0
2 -3 5
 
 =  2
 
1 0
-3 5
 
  - 3 
 
1 0
2 5
 
  +  (-1)
 
1 1
2 -3
 

=2(5+0)-3(5-0)-1(-3-2)

=10-15+5

=o

so A is singular

 iii) 
 
1 1 2 -1
1 2 -1 -3
2 3 1 2
3 -1 3 4
 

SOLUTION:

Let   A= 
 
1 1 2 -1
1 2 -1 -3
2 3 1 2
3 -1 3 4
 
lAl = 
 
1 1 2 -1
1 2 -1 -3
2 3 1 2
3 -1 3 4
 
      = 
 
1 0 0
1 1 -3 -2
2 1 -3 4
3 -4 -3 7
 
   C2-C1
  C3-2C1
  C4+C1
      = 
 
1 -3 -2
1 -3 4
-4 -3 7
 
  
 
 
      = 
 
1 0
1 0 6
-4 -15 -1
 
C2+3C1
C3+2C1
 =  1
 
0 6
-15 -1
 
  - 0 
 
1 6
-4 -1
 
  +  0
 
1 0
-4 -15
 

=1(0+90)

=90

≠0

So A is  non singular.

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