If A = [4, −1; 2, 1], find the value of k for which A² = kA − 6I.

1. 1 · Write what is given and what is to be found. Given: A = [4, −1; 2, 1] and A² = kA − 6I. To find: k.
2. 2 · Write the rule. Multiply row by column; equal matrices have equal corresponding entries.
3. 3 · Find A².
   • Row 1: (4)(4) + (−1)(2) = 14 and (4)(−1) + (−1)(1) = −5
   • Row 2: (2)(4) + (1)(2) = 10 and (2)(−1) + (1)(1) = −1
   • A² = [14, −5; 10, −1]
4. 4 · Find kA − 6I. kA − 6I = [4k − 6, −k; 2k, k − 6].
5. 5 · Equate the entries. Position (1, 2): −k = −5, so k = 5.
6. 6 · Check and box the answer. With k = 5: 4(5) − 6 = 14, 2(5) = 10 and 5 − 6 = −1. All four entries match, so k = 5.

Scoring tip: Find k from the simplest entry, then check it in all four positions. A mismatch means an arithmetic slip.