Let \(A\) be a matrix satisfying \(A^{3}=3A+2I\) where \(A \neq nI\), \(n \in \mathbb{Z}\). If \(A^{2}+\alpha A+\beta I=O\), then maximum value of \(|\alpha+\beta|\) is ________.
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Correct Answer: 3
Solution and Explanation
The minimal polynomial of \(A\) divides \((x-2)(x+1)^{2}\). It cannot be linear because that would make \(A=2I\) or \(A=-I\), i.e. \(A=nI\), which is excluded. Since \(A\) satisfies a quadratic \(A^{2}+\alpha A+\beta I=O\), the minimal polynomial is quadratic: either
(i) \((x-2)(x+1)=x^{2}-x-2\): \(\alpha=-1,\ \beta=-2\), \(|\alpha+\beta|=3\); or
(ii) \((x+1)^{2}=x^{2}+2x+1\): \(\alpha=2,\ \beta=1\), \(|\alpha+\beta|=3\).
In both cases \(|\alpha+\beta|=3\), so the maximum value is 3.
Answer: 3
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