आव्यूह `[(1,1,1),(1,-1,1),(2,1,-1)]` का प्रतिलोम ज्ञात कीजिये।

Explanation

माना `A=IA`
`implies [(1,1,1),(1,-1,1),(2,1,-1)]=[(1,0,0),(0,1,0),(0,0,1)]A`
`implies [(1,0,-2),(1,-1,1),(2,1,-1)]=[(-1,0,1),(0,1,0),(0,0,1)]A [R_(1) rarr R_(3)-R_(1)]`
`implies [(1,0,-2),(0,-1,3),(0,1,3)]=[(-1,0,1),(1,1,-1),(2,0,-1)]A[(R_(2) rarr R_(2)-R_(1)),(R_(3) rarr R_(3)-2R_(1))]`
`implies [(1,0,-2),(0,-1,3),(0,0,6)]=[(-1,0,0),(1,1,-1),(3,1,-2)]A [R_(3) rarr R_(3)+R_(2)]`
`implies [(1,0,-2),(0,-1,3),(0,0,1)]=[(-1,0,1),(1,1,-1),(1//2,1//6,-1//3)]A (R_(3) rarr 1/6 R_(3))`
`implies [(1,0,0),(0,-1,0),(0,0,1)]=[(0,1//3,1//3),(-1//2,1//2,0),(1//2,1//6,-1//3)]A (R_(1) rarr R_(1) +2R_(3), R_(2) rarr R_(2)-3R_(3))`
`implies [(1,0,0),(0,1,0),(0,0,1)]=[(0,1//3,1//3),(1//2,-1//2,0),(1//2,1//6,-1//3)]A (R_(2) rarr -R_(2))`
`:. A^(-1) =[(0,1//3,1//3),(1//2,-1//2,0),(1//2,1//6,-1//3)]`