Find the inverse of the matrix and use the inverse to solve

Find the inverse of the matrix:

and use the inverse to solve the linear equation for x:

Solution

Solution:

Inverse of A is given by:

A^-1 =adj A/det A

det A=1 det(1 0 1

-1 0 0

0 1 1)

=1 det (0 0 -0 det(-1 0 +1 det( -1 0

0 1) 0 1) 0 1)

det of 2*2 matrix (a b = ad-bc

c d)

=1(0-0)-0(-1-0)+1(-1-0)

=0-0-1

detA =-1

since detA not equal to zero

Inverse exists.

Adj A is given by transpose of cofactor matrix

cofactor matrix is

( det(0 0 -det(-1 0 +det(-1 0

1 1) 0 1) 0 1)

-det(0 1 +det(1 1 -det ( 1 0

1 1) 0 1) 0 1)

det(0 1 -det(1 1 +det( 1 0

0 0) -1 0) -1 0))

=(0 1 -1

1 1 -1

0 -1 0)

AdjA=take transpose of the above matrix.

= 0 1 0

1 1 -1

-1 -1 0)

therfore A^-1  =adjA/detA

=1/-1 ( 0 1 0

1 1 -1

-1 -1 0)

=(0/-1 1/-1 0/-1

1/-1 1/-1 -1\\-1

-1/-1 -1/-1 0/-1

A^-1 = ( 0 -1 0

-1 -1 1

1 1 0)

Given Ax=b

x=A^-1 b

order of A^-1 is 3*3 3 rows and 3 columns

order of b is 3*1 3 rows and one column.

since no of columns of A^-1 =3=no of rows of b

we can multiply two matrices

resultant matrix x order is 3*1 (3 rows and one column)

x=( 0 -1 0 (1

-1 -1 1 1

1 1 0) 1)

=(0*1+(-1)1+(0)1

-1(1) +(-1)1+(1)1

1(1)+1(1)+0(1) )

=( 0 -1 + 0

-1 -1 +1

1+1+0 )   

=(-1

-1

2)

so x=-1

y=-1

and z=2

is the solution set for the given system of equations.