Question

Find the inverse of the matrix,  `A=[[1,3,3],[1,4,3],[1,3,4]]`by using column transformations.

Answer 1

`A=[[1,3,3],[1,4,3],[1,3,4]]`

`|A|=|[1,3,3],[1,4,3],[1,3,4]|`

`=1(16-9)-3(4-3)+3(3-4)`

=1 ≠ 0

A^-1  Exists

consider A^-1A=I

`A^-1[[1,3,3],[1,4,3],[1,3,4]]=[[1,0,0],[0,1,0],[0,0,1]]`

Applying C₂ → C₂ -3C₁ and C₃→ C₃ - 3C₁,

`A^-1[[1,0,0],[1,1,0],[1,0,1]]=[[1,-3,-3],[0,1,0],[0,0,1]]`

Applying C₁→  C₁ - C₂,

`A^-1[[1,0,0],[0,1,0],[1,0,1]]=[[4,-3,-3],[-1,1,0],[0,0,1]]`

Applying C₃ → C₃ - C₁

`A^-1[[1,0,0],[0,1,0],[0,0,1]]=[[7,-3,-3],[-1,1,0],[-1,0,1]]`

`A^-1=[[7,-3,-3],[-1,1,0],[-1,0,1]]`