Cayley-Hamilton Theorem: Every square matrix satisfies its own characteristic equation $|P - \lambda I| = 0$. Alternatively, for smaller matrices, direct multiplication or checking the minimal polynomial is efficient.
Given $P = \begin{bmatrix} 2 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}$. We calculate $P^2$:
\[ P^2 = \begin{bmatrix} 2 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 2 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]
\[ P^2 = \begin{bmatrix} (2)(2)+(1)(-1) & (2)(1)+(1)(0) & 0 \\ (-1)(2)+(0)(-1) & (-1)(1)+(0)(0) & 0 \\ 0 & 0 & (1)(1) \end{bmatrix} \]
\[ P^2 = \begin{bmatrix} 3 & 2 & 0 \\ -2 & -1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]
Now, compare with $2P - I$:
\[ 2P - I = 2\begin{bmatrix} 2 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} - \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]
\[ 2P - I = \begin{bmatrix} 4 & 2 & 0 \\ -2 & 0 & 0 \\ 0 & 0 & 2 \end{bmatrix} - \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 3 & 2 & 0 \\ -2 & -1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]
Since $P^2 = 2P - I$ (Option A)
