Question

if `A = [(2,-1,1),(-1,2,-1),(1,-1,2)]` then find A⁻¹ by the adjoint method.

Answer 1

`A = [(2,-1,1),(-1,2,-1),(1,-1,2)]` 

For the given matrix A

`therefore A_11 = (-1)^(1+1)|(2,-1),(-1,2)|=3`

`therefore A_12 = (-1)^(1+2)|(-1,-1),(1,2)|=1`

`therefore A_13 = (-1)^(1+3)|(-1,2),(-1,-1)|=-1`

`therefore A_21 = (-1)^(2+1)|(-1,1),(-1,2)|=1`

`therefore A_22 = (-1)^(2+2)|(2,1),(1,2)|=3`

`therefore A_23 = (-1)^(2+3)|(2,-1),(1,-1)|=1`

`therefore A_31 = (-1)^(3+1)|(-1,1),(2,-1)|=-1`

`therefore A_32 = (-1)^(3+2)|(2,1),(-1,-1)|=1`

`therefore A_33 = (-1)^(3+3)|(2,-1),(-1,2)|=3`

`therefore "Co-factor matrix A"= [(3,1,-1),(1,3,1),(-1,1,3)]`

`"Adj A"= [(3,1,-1),(1,3,1),(-1,1,3)]`

Now, 

`|A| = |(2,-1,1),(-1,2,-1),(1,-1,2)|` 

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

= 6 - 1 - 1

= 4

 Now, `"A"^-1 = 1/|"A"| ("Adj"A)`

`therefore "A"^-1 = 1/4[(3,1,-1),(1,3,1),(-1,1,3)]`