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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.
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Solution
We know that, (AB)−1 = B−1A−1
B = `[(1,2,-2),(-1,3,0),(0,-2,1)]`
∴ |B| = 1 × 3 − 2 × (−1) − 2(2)
= 3 + 2 − 4
= 5 − 4
= 1
Now, A11 = 3, A12 = 1, A13 = 2
A21 = 2, A22 = 1, A23 = 2
A31 = 6, A32 = 2, A33 = 5
∴ adj B = `[(3,2,6),(1,1,2),(2,2,5)]`
Now, B−1 = `1/|B|` adj B
∴ B−1 = `[(3,2,6),(1,1,2),(2,2,5)]`
∴ (AB)−1 = B−1A−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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