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

$$A =[\begin{matrix}0 & -1 \\ 1 & 0\end{matrix}]$$A=[01​−10​]

$${A}^{2}= [\begin{matrix}0 & -1 \\ 1 & 0\end{matrix}][\begin{matrix}0 & -1 \\ 1 & 0\end{matrix}]= [\begin{matrix}-1 & 0 \\ 0 & -1\end{matrix}]$$A2=[01​−10​][01​−10​]=[−10​0−1​]

$${A}^{3}=[\begin{matrix}-1 & 0 \\ 0 & -1\end{matrix}][\begin{matrix}0 & -1 \\ 1 & 0\end{matrix}]=[\begin{matrix}0 & 1 \\ -1 & 0\end{matrix}]$$A3=[−10​0−1​][01​−10​]=[0−1​10​]

$${A}^{4}=[\begin{matrix}0 & 1 \\ -1 & 0\end{matrix}][\begin{matrix}0 & -1 \\ 1 & 0\end{matrix}]=[\begin{matrix}1 & 0 \\ 0 & 1\end{matrix}]$$A4=[0−1​10​][01​−10​]=[10​01​]

Option $(1)$(1) ${A}^{3}+I= A ({A}^{3}-I)$A3+I=A(A3−I)

$$[\begin{matrix}0 \\ 1\end{matrix} \begin{matrix}-1 \\ 0\end{matrix}] [\begin{matrix}-1 \\ -1\end{matrix} \begin{matrix}1 \\ -1\end{matrix}]= [\begin{matrix}1 \\ -1\end{matrix} \begin{matrix}1 \\ 1\end{matrix}]$$[01​−10​][−1−1​1−1​]=[1−1​11​]

$[\begin{matrix}1 \\ -1\end{matrix} \begin{matrix}1 \\ 1\end{matrix}]$[1−1​11​] = $[\begin{matrix}1 \\ -1\end{matrix} \begin{matrix}1 \\ 1\end{matrix}]$[1−1​11​] .....Correct

Option $(2)$(2) ${A}^{4}-I={A}^{2}+I$A4−I=A2+I

$[\begin{matrix}0 & 0 \\ 0 & 0\end{matrix}]= [\begin{matrix}0 & 0 \\ 0 & 0\end{matrix}]$[00​00​]=[00​00​] ....Correct

Option $(3)$(3) $[\begin{matrix}0 & 0 \\ 0 & 0\end{matrix}]= [\begin{matrix}0 & -1 \\ 1 & 0\end{matrix}]\times [\begin{matrix}-2 & 0 \\ 0 & -2\end{matrix}] = [\begin{matrix}0 & 2 \\ -2 & 0\end{matrix}]$[00​00​]=[01​−10​]×[−20​0−2​]=[0−2​20​] .....Incorrect

Option $(4)$(4)

$${A}^{3}-I=A(A-I)$$A3−I=A(A−I)

$$[\begin{matrix}-1 & 1 \\ -1 & -1\end{matrix}]= [\begin{matrix}0 & -1 \\ 1 & 0\end{matrix}] \times [\begin{matrix}-1 & -1 \\ 1 & -1\end{matrix}] = [\begin{matrix}-1 & 1 \\ -1 & -1\end{matrix}]$$[−1−1​1−1​]=[01​−10​]×[−11​−1−1​]=[−1−1​1−1​]

......Correct.