find the inverse of the matrix `[(1,2,5),(2,3,1),(-1,1,1)],` using elementary row operations.

Explanation

Let `A=[(1,2,5),(2,3,1),(-1,1,1)],`
`therefore" " |A|=[(1,2,5),(2,3,1),(-1,1,1)],=1(3-1)-2(2+1)+5(2+3)=21!=0`
`therefore" " A^(-1)` exists.
We write A=IA
`rArr" " [(1,2,5),(2,3,1),(-1,1,1)]=[(1,0,0),(0,1,0),(-0,0,1)]`
Applying `R_(2) to R_(2)` and `R_(3) to R_(3) +R_(1),` we get
` [(1,2,5),(0,-1,-9),(0,3,6)]=[(1,0,0),(-2,1,0),(1,0,1)]A`
Applying `R_(2) to (-1)R_(2)` and `R_(3) to ((1)/(3))R_(3),` we get ` [(1,2,5),(0,1,9),(0,1,2)]=[(1,0,0),(2,-1,0),((1)/(3),0,(1)/(3))]A`
Applying `R_(1) to R_(1)-2R_(2)` and `R_(3)to R_(3)-R_(2),` we get
` [(1,0,-13),(0,1,9),(0,0,-7)]=[(-3,2,0),(2,-1,0),((-5)/(3),1,(1)/(3))]A`
Applying `R_(3) to (-(1)/(7))` we get
` [(1,0,-13),(0,1,9),(0,0,1)]=[(-3,2,0),(2,-1,0),((-5)/(21),-(1)/(7),(1)/(21))]A`
Applying `R_(2) to R_(2)-9R_(3)` and `R_(1) to R_(1) +13R_(3),` we get
` [(1,0,0),(0,1,0),(0,0,1)]=[((2)/(21),(1)/(7),-(13)/(21)),(-(1)/(7),(2)/(7),(3)/(7)),((5)/(21),-(1)/(7),-(1)/(21))]A`
Hence `A^(-1)=[((2)/(21),(1)/(7),-(13)/(21)),(-(1)/(7),(2)/(7),(3)/(7)),((5)/(21),-(1)/(7),-(1)/(21))]`