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Question
Show that the matrices A = `[(2,2,1),(1,3,1),(1,2,2)]` and B = `[(4/5,(-2)/5,(-1)/5),((-1)/5,3/5,(-1)/5),((-1)/5,(-2)/5,4/5)]` are inverses of each other.
Solution
To prove that A and B are inverses of each other.
We have to prove that AB = BA = I.
Now AB = `[(2,2,1),(1,3,1),(1,2,2)][(4/5,(-2)/5,(-1)/5),((-1)/5,3/5,(-1)/5),((-1)/5,(-2)/5,4/5)]`
= `1/5[(2,2,1),(1,3,1),(1,2,2)][(4,-2,-1),(-1,3,-1),(-1,-2,4)]` ....[Take out 5 common from the matrix]
= `1/5[(8-2-1,-4+6-2,-2-2+4),(4-3-1,-2+9-2,-1-3+4),(4-2-2,-2+6-4,-1-2+8)] = 1/5[(5,0,0),(0,5,0),(0,0,5)]`
`= 1/5 xx 5[(1,0,0),(0,1,0),(0,0,1)] = [(1,0,0),(0,1,0),(0,0,1)]` = I
BA = `[(4/5,(-2)/5,(-1)/5),((-1)/5,3/5,(-1)/5),((-1)/5,(-2)/5,4/5)] [(2,2,1),(1,3,1),(1,2,2)]`
`= 1/5 [(4,-2,-1),(-1,3,-1),(-1,-2,4)] [(2,2,1),(1,3,1),(1,2,2)]`
`= 1/5[(8-2-1,8-6-2,4-2-2),(-2+3-1,-2+9-2,-1+3-2),(-2-2+4,-2-6+8,-1-2+8)]`
`= 1/5[(5,0,0),(0,5,0),(0,0,5)]`
`= 1/5 xx 5[(1,0,0),(0,1,0),(0,0,1)]`
`= [(1,0,0),(0,1,0),(0,0,1)]` = I
Thus AB = BA = I
Hence A and B are inverses of each other.
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