Question

What is the value of 'h' such that the matrix is augmented matrix of inconsistent system ?

Answer 1

There is no value of `h` for which this will be the augmented matrix of an inconsistent system `Ax=b`
(here I have assumed the standard convention that the first three columns stand for `A` and the last column is `b`)

Explanation:

The augmented matrix is

`((1,-1,1,|,3),(-1,8,2,|,h),(1,-2,-1,|,1))`

Carrying out row reduction we proceed as follows:

`R_2 -> R_2+R_1, R_3 -> R_3-R_1` :
`((1,-1,1,|,3),(0,7,3,|,h+3),(0,-1,-2,|,-2))`

`R_2 hArr R_3` (this is not necessary, just for simplification)
`((1,-1,1,|,3),(0,-1,-2,|,-2),(0,7,3,|,h+3))`

`R_2 -> -R_2`
`((1,-1,1,|,3),(0,1,2,|,2),(0,7,3,|,h+3))`

`R_3-7R_2`
`((1,-1,1,|,3),(0,1,2,|,2),(0,0,-11,|,h-11))`

Irrespective of the value of `h`, here both `A` and `(A|b)` have rank 3 - so this system is always consistent