If $A = \begin{bmatrix} 1 & 2 & 0 \\ 0 & 1 & 1 \\ 2 & 0 & 1 \end{bmatrix}$, show that $A^3 - 3A^2 + I = O$. Hence find $A^{-1}$.
Step-by-Step Solution
Key Concept: Compute A^2, A^3. A^{-1} = 3I - A^2 = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{bmatrix} - \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 4 & 1 \end{bmatrix} = \begin{bmatrix} 2 & -2 & -2 \\ -2 & 2 & -2 \\ -2 & -4 & 2 \end{bmatrix} / ... (or direct inverse derivation).
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Correct Answer: