Question

Show that AB = BA, where A = `[(-2, 3, -1),(-1, 2, -1),(-6, 9, -4)],"B" = [(1, 3, -1),(2, 2, -1),(3, 0, -1)]`.

Answer 1

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

= `[(-2 + 6 - 3, -6 + 6 - 0, 2 - 3 + 1),(-1 + 4 - 3, -3 + 4 - 0, 1 - 2 + 1),(-6 + 18 - 12, -18 + 18 + 0, 6 - 9 + 4)]`

∴ AB = `[(1, 0, 0),(0, 1, 0),(0, 0, 1)]`         ...(i)

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

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

∴ BA = `[(1, 0, 0),(0, 1, 0),(0, 0, 1)]`       ...(ii)

From (i) and (ii), we get
AB = BA.