Discussion
If a real matrix A has characteristic polynomial \((x - 1)(x^{2} + 1)\), then which of the following statements about A is true ?
Answer: D
A is invertible.
The determinant equals the product of the eigenvalues, the roots of \((x - 1)(x^{2} + 1)\): \(1 \cdot i \cdot (-i) = 1 \ne 0\). Equivalently, 0 is not a root of the characteristic polynomial, so A is invertible.
The others fail. A is not diagonalizable or triangulable over \(\mathbf{R}\), because both need the characteristic polynomial to split into real linear factors, and \(x^{2} + 1\) does not. A is not nilpotent, since a nilpotent matrix has characteristic polynomial \(x^{3}\).
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