Find the inverse of `{:((3,2,6),(1,1,2),(2,2,5)):}` by the adjoint method .

Explanation

`|A|=|{:(3,2,6),(1,1,2),(2,2,5):}|=3(5-4)-2(5-4)+6(2-2)`
`|A|=|{:(3,2,6),(1,1,2),(2,2,5):}|=3(5-4)-2(5-4)+6(2-2)`
`:.A^(-1)` exists.
`{:(M_(11)=5-4=1,",",A_(11)=(-1)^(2)xx1=1),(M_(12)=5-4=1,",",A_(12)=(-1)^(3)xx1=-1),(M_(13)=2-2=0,",",A_(13)=(-1)^(4)xx0=0),(M_(21)=10-12=-2,",",A_(21)=(-1)^(3)xx(-2)=2),(M_(22)=15-12=3,",",A_(22)=(-1)^(4)xx3=3),(M_(23)=6-4=2,",",A_(23)=(-1)^(5)xx2=-2),(M_(31)=4-6=-2,",",A_(31)=(-1)^(4)xx(-2)=-2),(M_(32)=6-6=0,",",A_(32)(-1)^(5)xx0=0),(M_(33)=3-2=1,",",A_(33)=(-1)^(6)xx1=1):}`
Hence, matrix of cofactors is
`[A_(ij)]_(3xx3)=[{:(A_(11),A_(12),A_(13)),(A_(21),A_(22),A_(23)),(A_(31),A_(32),A_(33)):}]`
`=[{:(1,-1,0),(2,3,-2),(-2,0,1):}]`
`:.adj.A=[A_(ij)]_(3xx3)^(T)=[{:(1,2,-2),(-1,3,0),(0,-2,1):}]`
Now `A^(-1)=(1)/(|A|).(adjA)`
`=(1)/(1).[{:(1,2,-2),(-1,3,0),(0,-2,1):}]`
`:.A^(-1)=[{:(1,2,-2),(-1,3,0),(0,-2,1):}]`