Question

Find the inverse of the following matrix using elementary operations.

`"A" = [(1,2,-2), (-1,3,0),(0,-2,1)]`

Answer 1

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)]`