Question

If A = `[(3,7),(2,5)]` and B = `[(6,8),(7,9)]`, then verify that (AB)^-1 = B^-1A^-1

Answer 1

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