Question

If A = `[(-1, 1, 1),(2, 3, 0),(1, -3, 1)],"B" = [(2, 1, 4),(3, 0, 2),(1, 2, 1)]`, state whether AB = BA? Justify your answer.

Answer 1

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

= `[(-2+ 3 + 1, -1 + 0 + 2, -4 + 2 + 1),(4 + 9 + 0, 2 + 0 + 0, 8 + 6 + 0),(2 - 9 + 1, 1 + 0 + 2, 4 - 6 + 1)]`

∴ AB = `[(2, 1, -1),(13, 2, 14),(-6, 3, -1)]`     ...(i)

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

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

∴ BA = `[(4, -7, 6),(-1, -3, 5),(4, 4, 2)]`      ...(ii)
From (i) and (ii), we get
AB ≠ BA.