आव्यूह `A=[(3,2,1),(0,-1,-2),(-3,4,2)]` से `A^(-1)` ज्ञात कीजिए।

Explanation

`|A|=|(3,2,1),(0,-1,-2),(-3,4,2)|`
`=3|(-1,-2),(4,2)|-2|(0,-2),(-3,2)|+1|(0,-1),(-3,4)|`
`=3(-2+8)-2(0-6)+1(0-3)`
`=18+12-3=27!=0`
`:.A` एक व्‍युत्‍क्रमणीय अव्‍यूह है।
अव्‍यूह A के सहगुणनखण्‍ड
`c_(11)=(-1)^(1+1)|(-1,-2),(4,2)|=(-2+8)=6`
`c_(12)=(-1)^(1+2)|(0,-2),(-3,2)|=-(0-6)=6`
`c_(13)=(-1)^(1+3)|(0,-1),(-3,4)|=0-3=-3`
`=c_(21)=(-1)^(2+1)+|(2,1),(4,2)|=-(4-4)=0`
`c_(22)=(-1)^(2+2)|(3,1),(-3,2)|=6+3=9`
`c_(23)=(-1)^(2+3)|(3,2),(-3,4)|=-(12+6)=-18`
`c_(31)=(-1)^(3+1)|(2,1),(-1,-2)|=-4+1=-3`
`c_(32)=(-1)^(3+2)|(3,1),(0,-2)|=-(-6-0)=6`
`c_(33)=(-1)^(3+3)|(3,2),(0,-1)|=-3-0=-3`
`:.adj.A=[(c_(11),c_(12),c_(13)),(c_(21),c_(22),c_(23)),(c_(31),c_(32),c_(33))]`
`[(6,6,-3),(0,9,-18),(-3,6,-3)]=[(6,0,-3),(6,9,6),(-3,-18,-3)]`
और `A^(-1)=1/(|A|).adj.A`
`=1/27[(6,0,-3),(6,9,6),(-3,18,-3)]=3/27[(2,0,-1),(2,3,2),(-1,-6,-1)]`
`=1/9[(2,0,-1),(2,3,2),(-1,-6,-1)]`