Fsc Part 1 Mathematics (Complete Solution)

# Q9 Find

(i)  A=

 3 2 -1 2 1 3

SOLUTION:

A =

 3 2 -1 2 1 3

At =

 3 2 2 1 -1 3

AAt =

 3 2 -1 2 1 3

 3 2 2 1 -1 3

=

 9+4+1 6+2-3 6+2-3 4+1+9

=

 14 5 5 14

lAAtl  =

 14 -5 5 14

=196-25

=171

AtA=

 3 2 2 1 -1 3

 3 2 -1 2 1 3

=

 9+4 6+2 -3+6 6+2 4+1 -2+3 -3+6 -2+3 1+9

=

 13 8 3 8 5 1 3 1 10

lAtAl  =

 13 8 3 8 5 1 3 1 10

=  13

 5 1 1 10

- 8

 8 1 3 10

+ 3

 8 5 3 1

= 13(50-1)-8(80-3)+3(8-15)

=636-616-21

=0

(ii)  A=

 3 2 2 1 1 1 2 3

SOLUTION:

A =

 3 2 2 1 1 1 2 3

At =

 3 2 1 2 4 1 1 3

AAt =

 3 2 2 1 1 1 2 3

 3 2 1 2 4 1 1 3

=

 9+16 6+4 3+4 6+12 6+4 4+1 2+1 4+3 3+4 2+1 1+1 2+3 6+12 4+3 2+3 4+9

lAAtl =

 25 10 7 18 10 5 3 7 7 3 2 5 8 7 5 13

=

 0 35 7 -7 0 15 3 -3 0 10 2 -2 1 -1 0 1

R1-25R4
R2-10R4
R3-7R4

Expand by C1

=-1

 35 7 -7 5 3 -3 0 2 -2

=-1

 35 7 -7 5 3 -3 0 2 -2

C2 and C3 are identical . so the value of determinant is equal to zero.

=0

=R.H.S

AtA=

 3 2 1 2 4 1 1 3

 3 2 2 1 1 1 2 3

=

 9+4+1+4 12+2+1+6 12+2+1+6 16+1+1+9

=

 18 21 21 27

lAtAl  =

 18 21 21 27

=(18)(27)-(21)(21)

=486-441

=45