Question

If A = `[(3, -2),(4, -2)]` and I = `[(1, 0),(0, 1)],` find k so that A² = kA – 2I

Answer 1

Given: A = `[(3, -2), (4,-2)],` I = `[(1,0),(0,1)]`

A² = A·A = `[(3, -2),(4, -2)] [(3, -2),(4, -2)]`

= `[(9 - 8, -6 + 4),(12 - 8, -8 + 4)]`

= `[(1, -2),(4, -4)]`

kA – 2I = k `[(3, -2),(4, -2)] - 2 [(1, 0),(0, 1)]`

= `[(3k, -2k),(4k, -2k)] - [(2, 0),(0, 2)]`

= `[(3k + 2, -2k),(4k, -2k + 2)]`

A² = kA – 2I

`[(1, -2),(4, -4)] = [(3k - 2, -2k),(4k, -2k - 2)]`

3k – 2 = 1

⇒ 3k = 3

k = 1