Question

Verify A(BC) = (AB)C, if A = `[(1, 0, 1),(2, 3, 0),(0, 4, 5)], "B" = [(2, -2),(-1, 1),(0, 3)] and "C" = [(3,2,-1), (2,0,-2)]`

Answer 1

BC = `[(2, -2),(-1, 1),(0, 3)][(3, 2, -1),(2, 0, -2)]`

= `[(6 - 4, 4 + 0, -2 + 4),(-3 + 2, -2 + 0, 1 - 2),(0 + 6, 0 + 0, 0 - 6)]`

= `[(2, 4, 2),(-1, -2, -1),(6, 0, -6)]`

∴ A(BC) = `[(1, 0, 1),(2, 3, 0),(0, 4, 5)][(2, 4, 2),(-1, -2, -1),(6, 0, -6)]`

= `[(2 + 0 + 6, 4 + 0 + 0, 2 + 0 - 6),(4 - 3 + 0, 8 - 6 + 0, 4 - 3 + 0),(0 - 4 + 30, 0 - 8 + 0, 0 - 4 - 30)]`

∴ A(BC) = `[(8, 4, -4),(1, 2, 1),(26, -8, -34)]`     ...(i)

AB = `[(1, 0, 1),(2, 3, 0),(0, 4, 5)][(2, -2),(-1, 1),(0, 3)]`

= `[(2 + 0 + 0, -2 + 0 + 3),(4 - 3 + 0, -4 + 3 + 0),(0 - 4 + 0, 0 + 4 + 15)]`

= `[(2, 1),(1, -1),(-4, 19)]`

∴ (AB)C = `[(2, 1),(1, -1),(-4, 19)][(3, 2, -1),(2, 0, -2)]`

= `[(6 + 2, 4 + 0, -2 - 2),(3 - 2, 2 + 0, -1 + 2),(-12 + 38, -8 + 0 , 4 - 38)]`

∴ (AB)C = `[(8, 4, -4),(1, 2, 1),(26, -8, -34)]`       ...(ii)

From (i) and (ii), we get

A(BC) = (AB)C