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Question
If `A^(-1) =[(3,-1,1),(-15,6,-5),(5,-2,2)]` and `B = [(1,2,-2),(-1,3,0),(0,-2,1)]` find `(AB)^(-1)`
Solution
We know that, `(AB)^-1 = B^-1 A^-1`.
B = `[(1,2,-2),(-1,3,0),(0,-2,1)]`
∴ |B| = `1 xx 3 - 2 xx (-1) - 2(2)`
= 3 + 2 - 4
= 5 - 4
= 1
Now,
`A_11 = 3, A_12 = 1, A_13 = 2`
`A_21 = 2, A_22 = 1, A_23 = 2`
`A_31 = 6, A_32 = 2, A_33 = 5`
∴ adjB = `[(3,2,6),(1,1,2),(2,2,5)]`
Now,
`B^-1 = 1/|B|.adjB`
∴ `B^-1 = [(3,2,6),(1,1,2),(2,2,5)]`
∴ `(AB)^-1 = B^-1 A^-1`
= `[(3,2,6),(1,1,2),(2,2,5)][(3,-1,1),(-15,6,-5),(5,-2,2)]`
= `[(9-30+30,-3+12-12,3-10+12),(3-15+10,-1+6-4,1-5+4),(6-30+25,-2+12-10,2-10+10)]`
= `[(9,-3,5),(-2,1,0),(1,0,2)]`
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