Question

If A = `[(1, 0, 2), (0, 2, 1), (2, 0, 3)]` and A³ – 6A² + 7A + kI = 0, find the value of k

Answer 1

A = `[(1, 0, 2), (0, 2, 1), (2, 0, 3)]`

A² = `"A" * "A"`

= `[(1, 0, 2), (0, 2, 1), (2, 0, 3)] [(1, 0, 2), (0, 2, 1), (2, 0, 3)]`

= `[(1 + 0 + 4, 0 + 0  0, 2 + 0 + 6),(0 + 0 + 2, 0 + 4 + 0, 0 + 2 + 3),(2 + 0 + 6, 0 + 0 + 0, 4 + 0 + 9)]`

A² = `[(5, 0, 8), (2, 4, 5), (8, 0, 13)]`

A³ = `"A"^2 * "A"`

= `[(5, 0, 8), (2, 4, 5), (8, 0, 13)] [(1, 0, 2), (0, 2, 1), (2, 0, 3)]`

A³ = `[(21, 0, 34),(12, 8, 23),(34, 0, 55)]`

A³ – 6A² + 7A + kI = 0

`[(21, 0, 34),(12, 8, 23),(34, 0, 55)] - 6[(5, 0, 8), (2, 4, 5),(8, 0, 13)] + 7[(1, 0, 2), (0, 2, 1), (2, 0, 3)] + "k"[(1, 0, 0), (0, 1, 0), (0, 0, 1)] = [(0, 0, 0), (0, 0, 0), (0, 0, 0)]`

`[(21, 0, 34),(12, 8, 23),(34, 0, 55)] - [(30, 0, 48), (12, 24, 30),(48, 0, 78)] + [(7, 0, 14), (0, 14, 7), (14, 0, 21)] + [("k", 0, 0), (0, "k", 0), (0, 0, "k")] = [(0, 0, 0), (0, 0, 0), (0, 0, 0)]`

`[(21 - 30 + 7 + "k", 0 - 0 + 0 + 0, 34 - 48 + 14 + 0), (12 - 12 + 0 + 0, 8 - 24 + 14 + "k", 23 - 30 + 7 + 0), (34 - 48 + 14 + 0, 0 - 0 + 0 + 0, 55 - 78 + 21 + "k")] = [(0, 0, 0), (0, 0, 0), (0, 0, 0)]`

`[(- 2 + "k", 0, 0), (0, - 2 - "k", 0), (0, 0, - 2 + "k")] = [(0, 0, 0), (0, 0, 0), (0, 0, 0)]`

Equating the corresponding entries

– 2 + k = 0

⇒ k = 2

∴ The required value of k is k = 2