Question

Using elementary row transformation, find the inverse of the matrix

`[(2,-3,5),(3,2,-4),(1,1,-2)]`

Answer 1

We have A =`[(2,-3,5),(3,2,-4),(1,1,-2)]`

A=IA `[(2,-3,5),(3,2,-4),(1,1,-2)]=[(1,0,0),(0,1,0),(1,0,0)]`A

R₁ ↔ R₃

`[(1,1,-2),(3,2,-4),(2,-3,5)]=[(0,0,1),(0,1,0),(1,0,0)]`A

R₂ → R₂ - 3R₁,
R₃ → R₃ - 2R₁

`[(1,1,-2),(0,-1,2),(0,-5,9)]=[(0,0,1),(0,1,-3),(1,0,-2)]`A

R₂ → R₁ +R_{2 ,}
R₃ → R₃ - 5R₂

`[(1,0,0),(0,-1,2),(0,0,-1)]=[(0,0,-2),(0,1,-3),(1,-5,13)]`A

R₂ → - R₂ ,
R₃ → - R₃

`[(1,0,0),(0,1,-2),(0,0,1)]=[(0,1,-2),(0,-1,3),(-1,5,-13)]`A

R₂ → R₂ + 2R₃

`[(1,0,0),(0,1,0),(0,0,1)]=[(0,1,-2),(-2,9,-23),(-1,5,-13)]`A

We know, I = AA^-1 

Therefore, inverse of A i.e. `"A"^-1 = [(0,1,-2),(-2,9,-23),(-1,5,-13)]`