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