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प्रश्न
If A = `[(3,7),(2,5)]` and B = `[(6,8),(7,9)]`, then verify that (AB)-1 = B-1A-1
उत्तर
Now AB = `[(3,7),(2,5)] [(6,8),(7,9)]`
`= [(18+49,24+63),(12+35,16+45)]`
`= [(67,87),(47,61)]`
|AB| = `= [(67,87),(47,61)]` = 4087 - 4089 = - 2
adj (AB) = `[(61,-87),(-47,67)]`
`("AB")^-1 = 1/|"AB"|` (adj AB)
`= 1/(-2) [(61,-87),(-47,67)]` ...(1)
Now we will find B-1A-1
B = `[(6,8),(7,9)]`, |B| = 54 - 56 = -2
adj B = `[(9,-8),(-7,6)]`
`"B"^-1 = 1/|"B"| ("adj B") = 1/(-2) [(9,-8),(-7,6)]`
A = `[(3,7),(2,5)]`, |A| = 15 - 14 = 1
adj A = `[(5,-7),(-2,3)]`
`"A"^-1 = 1/|"A"|` (adj A)
`= 1/1 [(5,-7),(-2,3)] = [(5,-7),(-2,3)]`
`"B"^-1"A"^-1 = 1/(-2)[(9,-8),(-7,6)][(5,-7),(-2,3)]`
`= 1/(-2)[(45+16,-63-24),(-35-12,49+18)]`
`= 1/(-2)[(61,-87),(-47,67)]` ......(2)
From (1) and (2),
(AB)-1 = B-1A-1
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