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Question
Find the inverse of the following matrix using elementary operations.
`"A" = [(1,2,-2), (-1,3,0),(0,-2,1)]`
Solution
We know that
A =I A
or, `[(1,2,-2),(-1,3,0),(0,-2,1)] =[(1,0,0),(0,1,0),(0,0,1)]"A"`
⇒ `[(1,2,-2),(0,5,-2),(0,-2,1)]=[(1,0,0),(1,1,0),(0,0,1)]"A"` ...[Applying R2 → R2 + R1]
⇒ `[(1,2,-2),(0,1,0),(0,-2,1)]=[(1,0,0),(1,1,2),(0,0,1)]"A"` ...[Applying R2 → R2 + 2R3]
⇒`[(1,0,-2),(0,1,0),(0,0,1)]=[(-1,-2,-4),(1,1,2),(2,2,5)]"A"
`...[Applying R1 → R1 + (-2)R2,R3 → R3 +2R2 ]
⇒`[(1,0,0),(0,1,0),(0,0,1)]=[(3,2,6),(1,1,2),(2,2,5)]"A"` ...[Applying R1→ R1+2R3 ]
Hence, `"A"^(-1) = [(3,2,6),(1,1,2),(2,2,5)]`
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