<p>(a) \(I\)</p>
<p>(b) \(2I\)</p>
<p>(c) \(\begin{bmatrix} 2 & -1 & 2 \\ -1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}\)</p>
<p>(d) \(\begin{bmatrix} 2 & 1 & -2 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}\)</p>
Step-by-Step Solution
Key Concept: Factor the characteristic polynomial to find which matrices satisfy the given matrix equation.
<p>The equation $A^3 - 2A^2 - A + 2I = 0$ can be factored as $(A - I)(A - 2I)(A + I) = 0$.</p><p>This means A must satisfy the equation if any eigenvalue is 1, 2, or -1.</p><p>Testing: (a) $A = I$: $I - 2I - I + 2I = 0$ ✓</p><p>(b) $A = 2I$: $8I - 8I - 2I + 2I = 0$ ✓</p><p>(c) and (d) can be verified by direct substitution into the matrix equation.</p>
Correct Answer: a, b, c, d