Discussion
Consider the matrix
\[ P = \begin{bmatrix} \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ 0 & 1 & 0 \\ \frac{-1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \end{bmatrix}. \]
Which one of the following statements about P is INCORRECT?
Answer: D
The incorrect statement is All eigenvalues of P are real numbers.
P is the matrix of a rotation by 45 degrees about the y axis. Its columns are orthonormal, so P is orthogonal (\(P^{T}P = I\)), its inverse equals its transpose, and its determinant is
\[ |P| = 1 \times \left(\tfrac{1}{2} + \tfrac{1}{2}\right) = 1 \]
A rotation keeps only its axis fixed, so it has one real eigenvalue, 1. The others satisfy \(\lambda^{2} - \sqrt{2}\lambda + 1 = 0\), giving \(\lambda = (1 \pm i)/\sqrt{2} = e^{\pm i\pi/4}\), which are complex. So the first three statements are true and the fourth is the incorrect one.
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