Fsc Part 1 Mathematics (Complete Solution)

Q7
Evaluate the following determinants:

(i)    
 
3 4 2 7
2 5 0 3
1 2 -3 5
4 1 -2 6
 
  
SOLUTION:  
 
3 4 2 7
2 5 0 3
1 2 -3 5
4 1 -2 6
 
  
      = 
 
0 -2 11 -8
0 1 6 -7
1 2 -3 5
0 -7 10 14
 
  R1-3R3
  R2-2R3
  R4-4R3

Expand by C1

      =1(-1)1+3 
 
-2 11 -8
1 6 -7
-7 10 -14
 
  R1-3R3
  R2-2R3
  R4-4R3
      = 
 
0 23 -22
1 6 -7
0 52 -63
 
  R1-2R2
  R3+7R2
      =-1 
 
23 -22
52 -63
 
  

=-[(-23)(63)+(22)(52)]

= -(-1449+1144)

= 305



(ii)    
 
2 3 1 -1
4 0 2 1
5 2 -1 6
3 -7 2 -2
 
  
SOLUTION:  
 
2 3 1 -1
4 0 2 1
5 2 -1 6
3 -7 2 -2
 
  
      = 
 
0 0 1 0
0 -6 2 3
7 5 -1 5
-1 -13 2 0
 
  C1-2C3
  C2-3C3
  C4-C3

Expand by R1

      =1(-1)1+3 
 
0 -6 2
7 5 5
-1 -13 0
 
  R1-3R3
  R2-2R3
  R4-4R3
      = 
 
0 0 3
7 15 5
-1 -13 0
 
  C2-2C3
      = 3 
 
7 15
-1 -13
 
  

=3(-91+15)

= 3(-76)

= 288



(iii)    
 
-3 9 1 1
0 3 -1 2
9 7 -1 1
-2 0 1 -1
 
  
SOLUTION:  
 
-3 9 1 1
0 3 -1 2
9 7 -1 1
-2 0 1 -1
 
  
      = 
 
0 0 0  1
6 -15 -3 2
12 -2 -2 1
-5 9 2 -1
 
  C1-3C4
  C2-3C4
  C3-C4

Expand by R1

      =1(-1)1+4 
 
6 -15 -3
12 -2 -2
-5 9 2
 
  
      =(-1)(3)(2) 
 
2 -5 -1
6 -1 -1
-5 9 2
 
  Taking Common 3 from R1
  and 2 from R2
      = -6 
 
0  0 -1
4 4 -1
-1 -1 0
 
  C1+2C3
  C2-5C3 
      = -6(-1)1+2 
 
4 4
-1 -1
 
   

=6(-4+4)

= 6(0)

= 0

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