निम्नलिखित समीकरण निकाय को आव्यूह विधि से हल कीजिए।
`x-y+2z=7,3x+4y-5z=-5,2x-y+3z=12`

Explanation

दिया समीकरण निकाय
`x-y+2z=7`
`3x+4y-5z=-5`
`2x-y-5z=-5`
`implies[(1,-1,2),(3,4,-5),(2,-1,3)][(x),(y),(z)]=[(7),(-5),(12)]impliesAX=B`………….1
`:.A=[(1,-1,2),(3,4,-5),(2,-1,3)]`
`implies|A|=|(1,-1,2),(3,4,-5),(2,-1,3)|`
`=1(12-5)-(-1)(9+10)+(-3-8)`
`=7+19-22=4!=0`
`:.A` व्‍युत्‍क्रमणीय है।
अब `A_(11)=7,A_(12=-19,A_(13)=-11`
`A_(21)=1,A_(22)=-1,A_(23)=-1`
`A_(31)=-3,A_(32)=11,A_(33)=7`
`:.adjA=[(7,-19,-11),(1,-1,-1),(-3,11,7)]=[(7,1,-3),(-19,-1,11),(-11,-1,7)]`
तथा `A^(-1)=1/(|A|).adjA=1/4[(7,1,-3),(-19,-1,11),(-11,-1,7)]`
समीकरण 1 से
`AX=BimpliesX=A^(-1)B`
`implies[(x),(y),(z)]=1/4[(7,1,-3),(-19,-1,11),(-11,-1,7)][(7),(-5),(12)]`
`=1/4[(49-5-36),(-133+5+132),(-77+5+84)]=[(2),(1),(3)]`
`:.x=-2,y-=1,z=3`