Question

Prove that (A + B) + C= A + (B + C), when A = `[(0, -1, 2),(3, 4, -5)], B = [(-2, 0, 3),(4, -5, 6)] and C = [(4, 7, -2),(0, -5, 1)]`

Answer 1

We are given three matrices:

A = `[(0, -1, 2),(3, 4, -5)], B = [(-2, 0, 3),(4, -5, 6)] and C = [(4, 7, -2),(0, -5, 1)]`

Find A + B

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

= `[(-2, -1, 5),(7, -1, 1)]`

Find (A + B) + C

(A + B) + C = `[(-2, -1, 5),(7, -1, 1)] + [(4, 7, -2),(0, -5, 1)]`

= `[(-2 + 4, -1 + 7, 5 + (-2)),(7 + 0, -1 + (-5), 1 + 1)]`

= `[(2, 6, 3),(7, -6, 2)]`

Find B + C

B + C = `[(-2 + 4, 0 + 7, 3 + (-2)),(4 + 0, -5 + (-5), 6 + 1)]`

= `[(2, 7, 1),(4, -10, 7)]`

Find A + (B + C)

A + (B + C) = `[(0, -1, 2),(3, 4, -5)] + [(2, 7, 1),(4, -10, 7)]`

= `[(0 + 2, -1 + 7, 2 + 1),(3 + 4, 4 + (-10), -5 + 7)]`

= `[(2, 6, 3),(7, -6, 2)]`

Compare:

(A + B) + C = `[(2, 6, 3),(7, -6, 2)]`

A + (B + C) = `[(2, 6, 3),(7, -6, 2)]`

∴ (A + B) + C = A + (B + C)

Hence proved.