आव्यूह `[{:(0,1,2),(1,2,3),(3,1,1):}]` का प्रतिलोम ज्ञात कीजिए ।

Explanation

माना `A=[{:(0,1,2),(1,2,3),(3,1,1):}]=[{:(a_(11),a_(12),a_(13)),(a_(21),a_(22),a_(23)),(a_(31),a_(32),a_(33)):}]`
`|A|=|{:(0,1,2),(1,2,3),(3,1,1):}|`
`=0(2-3)-1(1-9)+2(1-6)=-2ne0` इस प्रकार `A^(-1)` का अस्तित्व है। `A_(11)=a_(11)`
अर्थात 0 का सहखण्ड `=(-1)^(1+1)|{:(2,3),(1,1):}|=2-3=-1`
`A_(12)` अर्थात 1 का सहखण्ड
`=(-1)^(1+2)|{:(1,3),(3,1):}|=-(1-9)=8`
`A_(13)=(-1)^(1+3)|{:(1,2),(3,1):}|=1-6=-5`
`A_(21)=(-1)^(2+1)|{:(1,2),(1,1):}|=-(1-2)=1`
`A_(22)=(-1)^(2+2)|{:(0,2),(3,1):}|=0-6=-6`
`A_(23)=(-1)^(2+3)|{:(0,1),(3,1):}|=-(0-3)=3`
`A_(31)=(-1)^(3+1)|{:(1,2),(2,3):}|=3-4=-1`
`A_(32)=(-1)^(3+2)|{:(0,2),(1,3):}|=-(0-2)=2`
`A_(33)=(-1)^(3+3)|{:(0,1),(1,2):}|=0-1=-1`
माना `B=[{:(A_(11),A_(12),A_(13)),(A_(21),A_(22),A_(23)),(A_(31),A_(32),A_(33)):}]=[{:(-1,8,-5),(1,-6,3),(-1,2,-1):}]`
`adjA=B^(T)=[{:(-1,1,-1),(8,-6,2),(-5,3,-1):}]`
`A^(-1)=(adjA)/(|A|)=(adjA)/(-2)=[{:((1)/(2),-(1)/(2),(1)/(2)),(-4,3,-1),((5)/(2),-(3)/(2),(1)/(2)):}]`