Fsc Part 1 Mathematics (Complete Solution)

# Q9 Find the inverses of the following matrices. Also find their inverses by using row and column operations.

(i)

 1 2 -3 0 -2 0 -2 -2 2

SOLUTION:

lAl=

 1 2 -3 0 -2 0 -2 -2 2

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

=-4+0+12

=8

Cofactors of A =

 A11 A12 A13 A21 A22 A23 A31 A32 A33

Aij=(-1)i+jMij

A11  =  (-1)1+1

 -2 0 2 2

=(-1)2 (-4-0) =-4
A12  =  (-1)1+2

 0 0 -2 2

=(-1)3 (0+0) =0
A13  =  (-1)1+3

 0 -2 -2 -2

=(-1)4 (0-4) =-4
A21  =  (-1)2+1

 2 -3 -2 2

=(-1)3 (4-6) =2
A22  =  (-1)2+2

 1 -3 -2 2

=(-1)4 (2-6) =-4
A23  =  (-1)2+3

 1 2 -2 -2

=(-1)5 (-2+4) =-2
A31  =  (-1)3+1

 2 -3 -2 0

=(-1)4 (0-6) =-6
A32  =  (-1)3+2

 1 -3 0 0

=(-1)5 (0+0) =0
A33  =  (-1)3+3

 1 2 0 -2

=(-1)6 (-2-0) =-2
Cofactors of A =

 -4 0 -4 2 -4 -2 -6 0 -2

 -4 2 -4 0 -4 0 -4 -2 -2

SO,  A-1=
=
 1 8

 -4 2 -4 0 -4 0 -4 -2 -2

=

 -1/2 1/4 -3/4 0 -1/2 0 -1/2 -1/4 -1/4

(ii)

 1 2 -2 0 -1 3 1 0 2

SOLUTION:

lAl=

 1 2 -1 0 -1 3 1 0 2

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

=1(-2)-2(-3)-1(1)

=-2+6-1

=3

Cofactors of A =

 A11 A12 A13 A21 A22 A23 A31 A32 A33

Aij=(-1)i+jMij

A11  =  (-1)1+1

 -1 3 0 2

=(-1)2 (-2-0) =-2
A12  =  (-1)1+2

 0 3 1 2

=(-1)3 (0-3) =-3
A13  =  (-1)1+3

 0 -1 1 0

=(-1)4 (0+1) =1
A21  =  (-1)2+1

 2 -1 0 2

=(-1)3 (4+0) =4
A22  =  (-1)2+2

 1 -1 1 2

=(-1)4 (2+1) =3
A23  =  (-1)2+3

 1 2 1 0

=(-1)5 (0-2) =2
A31  =  (-1)3+1

 2 -1 -1 3

=(-1)4 (6-1) =5
A32  =  (-1)3+2

 1 -1 0 3

=(-1)5 (3+0) =-3
A33  =  (-1)3+3

 1 2 0 -1

=(-1)6 (-1-0) =-1
Cofactors of A =

 -2 3 1 -2 3 2 5 -3 -1

 -2 -4 5 3 3 -3 1 2 -1

SO,  A-1=
=
 1 3

 -2 -4 5 3 3 -3 1 2 -1

=

 -2/3 -4/3 5/3 1 1 -1 1/3 2/3 -1/3

(iii)

 1 -3 2 2 1 0 0 -1 1

SOLUTION:

lAl=

 1 -3 2 2 1 0 0 -1 1

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

=1+6-4

=3

Cofactors of A =

 A11 A12 A13 A21 A22 A23 A31 A32 A33

Aij=(-1)i+jMij

A11  =  (-1)1+1

 1 0 -1 1

=(-1)2 (1-0) =1
A12  =  (-1)1+2

 2 0 0 1

=(-1)3 (2-0) =-2
A13  =  (-1)1+3

 2 1 0 -1

=(-1)4 (-2-0) =-2
A21  =  (-1)2+1

 -3 2 -1 1

=(-1)3 (-3+2) =1
A22  =  (-1)2+2

 1 2 0 1

=(-1)4 (1-0) =1
A23  =  (-1)2+3

 1 2 1 0

=(-1)5 (-1-0) =1
A31  =  (-1)3+1

 -3 2 1 0

=(-1)4 (0-2) =-2
A32  =  (-1)3+2

 1 2 2 0

=(-1)5 (0-4) =4
A33  =  (-1)3+3

 1 -3 2 1

=(-1)6 (1+6) =7
Cofactors of A =

 1 -2 -2 1 1 1 -2 4 7

 1 1 -2 -2 1 4 -2 1 7

SO,  A-1=
=
 1 3

 1 1 -2 -2 1 4 -2 1 7

=

 1/3 1/3 -2/3 -2/3 1/3 4/3 -2/3 1/3 7/3