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