If A = [2, 1; −1, 3] and B = [1, 0; 2, −1], verify that (AB)′ = B′A′.

1. 1 · Write what is given and what is to be found. Given: A = [2, 1; −1, 3] and B = [1, 0; 2, −1]. To verify: (AB)′ = B′A′.
2. 2 · Write the rule. Multiply row by column; the transpose turns rows into columns.
3. 3 · Find AB and (AB)′ (the left side).
   • AB = [(2)(1) + (1)(2), (2)(0) + (1)(−1); (−1)(1) + (3)(2), (−1)(0) + (3)(−1)] = [4, −1; 5, −3]
   • (AB)′ = [4, 5; −1, −3]
4. 4 · Find B′ and A′. B′ = [1, 2; 0, −1] and A′ = [2, −1; 1, 3].
5. 5 · Find B′A′ (the right side). B′A′ = [(1)(2) + (2)(1), (1)(−1) + (2)(3); (0)(2) + (−1)(1), (0)(−1) + (−1)(3)] = [4, 5; −1, −3].
6. 6 · Check and box the answer. LHS = RHS = [4, 5; −1, −3]. Hence (AB)′ = B′A′ is verified.

Scoring tip: Label the left-hand side and the right-hand side clearly, then write 'LHS = RHS'. Examiners look for that closing line.