If `A=[1 2 0-2-1-2 0-1 1]` , find `A^(-1)` . Using `A^(-1)` , solve the system of linear equations `x-2y=10 ,\ \ 2x-y-z=8,\ \ -2y+z=7`

Answer

N/a

Explanation

We have `A=|(1,2,0),(-2,-1,-2),(0,-1,1)|`……………….(i)
`:. |A|=1(-3)-2(-2)+0=1!=0`
Now, `A_(11)=-3,A_(12)=2,A_(13)=2,A_(21)=-2,A_(22)=1,A_(23)=1,A_(31)=-4,A_(32)=2` and `A_(33)=3`
`:.` adj `(A)=|(-3,2,2),(-2,1,1),(-4,2,3)|^(T)=|(-3,-2,-4),(2,1,2),(2,1,3)|`
`A^(-1)=(adjA)/(|A|)`
`=1/1|(-3,-2,-4),(2,1,2),(2,1,3)|`
`implies A^(-1)=|(-3,-2,-4),(2,1,2),(2,1,3)|`................(ii)
Also we have the system of linear equations as
`x-2y=10`,
`2x-y-z=8`
and `-2y+z=7`
In the form `CX=D`
`[(1,2,0),(2,-1,-1),(0,-2,1)][(x),(y),(z)]=[(10),(8),(7)]`
Where `C=[(1,-2,0),(2,-1,-1),(0,-2,1)],X=[(x),(y),(z)]` and `D=[(10),(8),(7)]`
We know that `(A^(T))^(-1)=(A^(-1))^(T)`
`:.C^(T)=|(1,2,0),(-2,-1,-2),(0,-1,1)|=A` [using Eq. (i)]
`:. X=C^(-1)D`
`implies [(x),(y),(z)]=[(-3,2,2),(-2,1,1),(-4,2,3)][(10),(8),(7)]`
`=[(-30+16+14),(-20+8+7),(-40+16+24)]=[(0),(-5),(-3)]`
`:. x=0, y=-5` and `z=-3`