निम्नलिखित समीकरण निकाय को आव्यूह विधि से हल कीजिए।
`x-y+z=4,2x+y-3z=0,x+y+z=2`

Explanation

दिया समीकरण निकाय
`x+y+z=4`
`2x+y-3z=0`
`x+y+z=2`
`implies[(1,-1,1),(2,1,-3),(1,1,1)][(x),(y),(z)]=[(4),(0),(2)]impliesAX=B`…………1
`:.A=[(1,-1,1),(2,1,-3),(1,1,1)]`
`implies|A|=|(1,-1,1),(2,1,-3),(1,1,1)|`
`=1(1+3)-(-1)(2+3)+1(2-1)`
`=4+5+1=10!=0`
`:.A` व्‍तुत्‍क्रमणीय है।
अब `A_(11)-4,A_(12)=-5,A_(13)=1`
`A_(21)=2,A_(22)=0,A_(23)=-2`
`A_(31)=2,A_(32)=5,A_(33)=3`
`:.adjA=[(4,-5,1),(2,0,-2),(2,5,3)]=[(4,2,2),(-5,0,5),(1,-2,3)]`
तथा `A^(-1)1/(|A|)-adjA=1/10[(4,2,2),(-5,0,5),(1,-2,3)]`
समीकरण 1 से
`AX=BimpliesX=A^(-1)B`
`implies[(x),(y),(z)]=1/10[(4,2,2),(-5,0,5),(1,-2,3)][(4),(0),(2)]`
`=1/10[(16+0+4),(-20+2+10),(4+0+6)]=[(2),(-1),(1)]`
`:.x=2,=-1,z=1`