Discussion
If A is a real 3 × 3 matrix, which of the following statements can not be true about A ?
Answer: D
The impossible statement is that the minimum polynomial of A is \(x^{2} + 1\).
The characteristic polynomial of a real 3 × 3 matrix has real coefficients and odd degree, so it has a real root \(\lambda\). That real eigenvalue must also be a root of the minimal polynomial. But \(x^{2} + 1\) has no real roots, so it cannot be the minimal polynomial.
The others can happen. A companion matrix has minimal polynomial equal to its characteristic polynomial (A). Every real eigenvalue is also a complex number, so A always has a complex eigenvalue (B). And \(\mathrm{diag}(1, -1, 1)\) has minimal polynomial \(x^{2} - 1\) (C).
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