If $$A = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$$, then which one of the following statements is not correct?

Given matrix $$ A = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} $$, we need to determine which statement among the options is not correct. First, we compute the powers of $$ A $$ to use in the verification.

Calculate $$ A^2 $$:

$$ A^2 = A \times A = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} 0 \cdot 0 + (-1) \cdot 1 & 0 \cdot (-1) + (-1) \cdot 0 \\ 1 \cdot 0 + 0 \cdot 1 & 1 \cdot (-1) + 0 \cdot 0 \end{bmatrix} = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} = -I $$

So, $$ A^2 = -I $$.

Calculate $$ A^3 $$:

$$ A^3 = A^2 \times A = (-I) \times A = -A $$

So, $$ A^3 = -A $$.

Calculate $$ A^4 $$:

$$ A^4 = A^3 \times A = (-A) \times A = -A^2 = -(-I) = I $$

So, $$ A^4 = I $$.

Now, we have:

$$ A^2 = -I, \quad A^3 = -A, \quad A^4 = I $$

We will check each option by substituting these values.

Option A: $$ A^3 + I = A(A^3 - I) $$

Left side: $$ A^3 + I = -A + I = I - A $$

Right side: $$ A(A^3 - I) = A(-A - I) = A \times (-A) + A \times (-I) = -A^2 - A = -(-I) - A = I - A $$

Since both sides equal $$ I - A $$, option A is correct.

Option B: $$ A^4 - I = A^2 + I $$

Left side: $$ A^4 - I = I - I = 0 $$ (the zero matrix)

Right side: $$ A^2 + I = -I + I = 0 $$

Since both sides equal the zero matrix, option B is correct.

Option C: $$ A^2 + I = A(A^2 - I) $$

Left side: $$ A^2 + I = -I + I = 0 $$

Right side: $$ A(A^2 - I) = A(-I - I) = A(-2I) = -2A = -2 \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 2 \\ -2 & 0 \end{bmatrix} $$

Left side is $$ 0 $$ (zero matrix) and right side is $$ \begin{bmatrix} 0 & 2 \\ -2 & 0 \end{bmatrix} $$, which are not equal. Thus, option C is not correct.

Option D: $$ A^3 - I = A(A - I) $$

Left side: $$ A^3 - I = -A - I $$

Right side: $$ A(A - I) = A \times A - A \times I = A^2 - A = -I - A $$

Since both sides equal $$ -I - A $$, option D is correct.

Hence, the statement that is not correct is in Option C. So, the answer is Option C.