Question

If A = `[(2, 3), (1, 2)], B = [(1, 0),(3, 1)]`, Find (AB)^-1

Answer 1

A = `[(2, 3), (1, 2)] , B = [(1, 0),(3, 1)]`

AB = `[(2, 3), (1, 2)] [(1, 0),(3, 1)] = [(11, 3),(7, 2)]`

`|AB| = [(11, 3),(7, 2)] = 22 - 21 = 1 ≠ 0`

To find (AB)^-1

(AB).(AB)^-1

`[(11, 3),(7, 2)](AB)^-1 = [(1, 0), (0, 1)]`

`R_1 -> (1/11)R`

`[(1, 3/11), (7, 2)](AB)^-1 = [(1/11, 0), (0, 1)]`

`R_2 -> R_2 - 7R_1`

`[(1, 3/11), (0, 1/11)](AB)^-1 = [(1/11, 0), (-7/11, 1)]`

`R_1 -> R_1 - 3R_2`

`[(1, 0), (0, 1/11)](AB)^-1  = [(2, -3), (-7/11, 1)]`

`R_2 -> (11)R_2`

`[(1, 0), (0, 1)](AB)^-1 = [(2, -3),(-7, 11)]`

`therefore (AB)^-1 =  [(2, -3),(-7, 11)]`