Question

If A = `[(1, 2),(2, 3)]` and A² – kA – I₂ = 0, then value of k is ______.

Options

• 4

• 2

• 1

• –4

Answer 1

If A = `[(1, 2),(2, 3)]` and A² – kA – I₂ = 0, then value of k is 4.

Explanation:

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

A² = `[(1, 2),(2, 3)][(1, 2),(2, 3)]`

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

= `[(5, 8),(8, 13)]`

A² – kA – I₂ = 0

⇒ `[(5, 8),(8, 13)] - k[(1, 2),(2, 3)] - [(1, 0),(0, 1)]` = 0

⇒ `[(5 - k - 1, 8 - 2k - 0),(8 - 2k - 0, 13 - 3k - 1)]` = 0

⇒ `[(4 - k, 8 - 2k),(8 - 2k, 12 - 3k)]` = 0

= `[(0, 0),(0, 0)]`

⇒ k = 4