If A = [1, 2; 3, 4] and B = [2, 0; 1, 3], find AB and verify that (AB)ᵀ = BᵀAᵀ.

1. 1 · Write what is given and what is to be found. A and B are both of order 2 × 2. To find: AB, then show that (AB)ᵀ = BᵀAᵀ.
2. 2 · Write the rule. Element (i, j) of AB = row i of A × column j of B. The property to verify is (AB)ᵀ = BᵀAᵀ.
3. 3 · Find AB.
   • Row 1 × column 1: 1 × 2 + 2 × 1 = 4
   • Row 1 × column 2: 1 × 0 + 2 × 3 = 6
   • Row 2 × column 1: 3 × 2 + 4 × 1 = 10
   • Row 2 × column 2: 3 × 0 + 4 × 3 = 12
   • So AB = [4, 6; 10, 12]
4. 4 · Work out the left side. LHS = (AB)ᵀ = [4, 10; 6, 12].
5. 5 · Work out the right side. Bᵀ = [2, 1; 0, 3] and Aᵀ = [1, 3; 2, 4], so RHS = BᵀAᵀ = [4, 10; 6, 12].
6. 6 · Check and box the answer. LHS = RHS = [4, 10; 6, 12]. Hence verified.

Scoring tip: Show the row × column working for every element. Label LHS and RHS clearly, and end with 'Hence verified'.