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आव्यूह विधि से निम्नलिखित समीकरण निकाय हो हल कीजिएः
`4x+3y+2z=60`
`x+2y+3z=45`
`6x+2y+3z=70.`

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निकाय गए समीकरण निकाय का आव्यूह रूप है
`AX=b.`
जहाँ `A=[{:(4,3,2),(1,2,3),(6,2,3):}], X =[{:(x),(y),(z):}],B=[{:(60),(45),(70):}]`
अब `|A| =|{:(4,3,2),(1,2,3),(6,2,3):}|`
`Rightarrow" "|A|= 4 |{:(2,3),(2,3):}|-3xx|{:(1,3),(6,3):}|+2|{:(1,2),(6,2):}|`
`Rightarrow" "|A| =4xx0-3xx(-15)+2(10)=25`
`Rightarrow" "|A| !=0` A अर्थार्त A व्यक्रामणिये है
अंत: समीकरण निकाय संगत है और अद्वितीय हल निम्न है
`X=A_(-1)B " "...(1)`
माना A में `a_(ij)` अवयव का सहखण्ड `A_(ij)` है तब
`A_(ij)` `A_11=(-1)^(1+1) |{:(,2,3),(,2,3):}|=1(6-6)=0`
`A_12=(-1)^(1+2) |{:(,1,3),(,6,3):}|= -(3-18)= 15`
`A_13=(-1)^(1+3) |{:(,1,2),(,6,2):}|= 1(12-12)= -10`
`A_21=(-1)^(2+1) |{:(,3,2),(,2,3):}|= -(9-4)= -5`
`A_22=(-1)^(2+2) |{:(,4,2),(,6,3):}|= 12-12= 0`
`A_23=(-1)^(2+3) |{:(,4,3),(,6,2):}|= -(8-18)= 10`
`A_31=(-1)^(3+1) |{:(,3,2),(,2,3):}|= 9-4= 5`
`A_32=(-1)^(3+2) |{:(,4,2),(,1,3):}|= -(12-2)= -10`
`A_33=(-1)^(3+3) |{:(,4,3),(,1,2):}|= 8-3= 5`
`:. adj A =[{:(0,15,-10),(-5,0,10),(5,-10,5):}]^(,)`
`=[{:(0,-5,5),(15,0,10),(-10,10,5):}]`
`:." "A^(-1) =(adj A)/(|A|) =(1)/(25)[{:(0,-5,5),(15,0,10),(-10,10,5):}]`
(1)
`[{:(x),(y),(z):}](1)/(25)[{:(0,-5,5),(15,0,10),(-10,10,5):}][{:(60),(45),(70):}]`
`Rightarrow" "[{:(x),(y),(z):}](1)/(25)[{:(0-225+350),(900+0-700),(-600+450+350):}]`
`[{:(x),(y),(z):}]=(1)/(25)[{:(125),(200),(200):}]`
`Rightarrow" "` x=5, y=8 और z=8.

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