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For the matrix A = 1 1 0 1 2 1 2 1 0 , which of the following options is correct?

Options

  1. AA 3 + 3A 2 - I = 0
  2. BA 3 - 3A 2 - I = 0
  3. CA 3 + 2A 2 - I = 0
  4. DA 3 - A 2 + I = 0

Correct answer

B. A 3 - 3A 2 - I = 0

Step-by-step solution

Given matrix is A = 1 1 0 1 2 1 2 1 0 ⇒ A 2 = A × A = 1 1 0 1 2 1 2 1 0 × 1 1 0 1 2 1 2 1 0 = 2 3 1 5 6 2 3 4 1 ⇒ A 3 = A 2 × A = 2 3 1 5 6 2 3 4 1 × 1 1 0 1 2 1 2 1 0 = 7 9 3 1 5 1 9 6 9 1 2 4 ∴ A 3 - I = 7 9 3 1 5 1 9 6 9 1 2 4 - 1 0 0 0 1 0 0 0 1 = 6 9 3 1 5 1 8 6 9 1 2 3 = 3 × 2 3 1 5 6 2 3 4 1 = 3 A 2 ⇒ A 3 - 3 A 2 - I = 0

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