If A = [4, 1; 3, 1], show that A² − 5A + I = O. Hence find A⁻¹.

1. 1 · Write what is given and what is to be found. Given: A = [4, 1; 3, 1]. To show: A² − 5A + I = O. To find: A⁻¹.
2. 2 · Write the rule. Multiply row by column. A matrix B is A⁻¹ if AB = BA = I.
3. 3 · Find A². A² = [(4)(4) + (1)(3), (4)(1) + (1)(1); (3)(4) + (1)(3), (3)(1) + (1)(1)] = [19, 5; 15, 4].
4. 4 · Show the equation. A² − 5A + I = [19 − 20 + 1, 5 − 5 + 0; 15 − 15 + 0, 4 − 5 + 1] = [0, 0; 0, 0] = O.
5. 5 · Rearrange to find A⁻¹.
   • 5A − A² = I
   • So A(5I − A) = I and (5I − A)A = I.
   • A⁻¹ = 5I − A = [5 − 4, 0 − 1; 0 − 3, 5 − 1] = [1, −1; −3, 4]
6. 6 · Check and box the answer. A × [1, −1; −3, 4] = [4 − 3, −4 + 4; 3 − 3, −3 + 4] = I. So A⁻¹ = [1, −1; −3, 4].

Scoring tip: Show that the product works in both orders, or say that 5I − A commutes with A. Then the inverse is fully justified.