If A = [[0, 1], [1, 0]], then which one of the following statements is not correct?

A = [[0, 1], [1, 0]]

A² = [[0, 1], [1, 0]] [[0, 1], [1, 0]] = [[1, 0], [0, 1]] = I

A³ = A²A = IA = A = [[0, 1], [1, 0]]

A⁴ = A²A² = IA = I = [[1, 0], [0, 1]]

Let's check each option:

1. A³ + I = A(A³ - I)
   A + I = A(A - I)
   [[0, 1], [1, 0]] + [[1, 0], [0, 1]] = [[0, 1], [1, 0]]([[0, 1], [1, 0]] - [[1, 0], [0, 1]])
   [[1, 1], [1, 1]] = [[0, 1], [1, 0]][[-1, 1], [1, -1]]
   [[1, 1], [1, 1]] = [[-1, -1], [-1, -1]] (Incorrect)

2. A³ - I = A(A - I)
   A - I = A(A - I)
   [[0, 1], [1, 0]] - [[1, 0], [0, 1]] = [[0, 1], [1, 0]]([[0, 1], [1, 0]] - [[1, 0], [0, 1]])
   [[-1, 1], [1, -1]] = [[0, 1], [1, 0]][[-1, 1], [1, -1]]
   [[-1, 1], [1, -1]] = [[-1, 1], [1, -1]] (Correct)

3. A⁴ - I = A² + I
   I - I = I + I
   [[0, 0], [0, 0]] = [[2, 0], [0, 2]] (Incorrect)

4. A² + I = A(A² - I)
   I + I = A(I - I)
   2I = A(0) = [[0, 0], [0, 0]] (Incorrect)

Therefore, the statement A³ + I = A(A³ - I) is not correct.