Find adj A for A = [1, −1, 2; 0, 2, −3; 3, −2, 5] and verify that A(adj A) = |A| I.

1. 1 · Write what is given and what is to be found. Given: A, a 3 × 3 matrix. To find: adj A, and show A(adj A) = |A| I.
2. 2 · Write the rule. Aᵢⱼ = (−1)ⁱ⁺ʲ Mᵢⱼ; adj A = transpose of [Aᵢⱼ]; A(adj A) = |A| I.
3. 3 · Find the nine cofactors.
   • A₁₁ = 4, A₁₂ = −9, A₁₃ = −6
   • A₂₁ = 1, A₂₂ = −1, A₂₃ = −1
   • A₃₁ = −1, A₃₂ = 3, A₃₃ = 2
4. 4 · Transpose to get adj A. adj A = [4, 1, −1; −9, −1, 3; −6, −1, 2].
5. 5 · Find |A|. |A| = 1(4) + (−1)(−9) + 2(−6) = 4 + 9 − 12 = 1.
6. 6 · Multiply A by adj A. Row 1 of A gives 4 + 9 − 12 = 1, 1 + 1 − 2 = 0 and −1 − 3 + 4 = 0. Rows 2 and 3 give 0, 1, 0 and 0, 0, 1 in the same way.
7. 7 · Check and box the answer. A(adj A) = I = 1 × I = |A| I. Verified.

Scoring tip: Show at least one full row of the product A(adj A). Writing only 'verified' earns no marks.