Correct answer: (D) All eigenvalues of P are real.
Explanation
The matrix is a rotation by $45^\circ$ about the y-axis. Its columns are orthonormal, so $P^TP=I$ and $P^{-1}=P^T$. Its determinant is 1. The fixed y direction has eigenvalue 1, while the x–z rotation has eigenvalues $e^{i\pi/4}$ and $e^{-i\pi/4}$, which are not real. Thus D is the incorrect statement.
More Linear Algebra: Matrix algebra, systems of linear equations, eigen values and eigen vectors questions
GATE 2025 ME Q11– Let **A** and **B** be real symmetric matrices of same size. Which one of the following…
GATE 2024 ME Q14– Consider the system of linear equations x + 2y + z = 5 2x + ay + 4z = 12 2x + 4y + 6z = b…
GATE 2024 ME Q36– The matrix bmatrix 1 & a \\ 8 & 3 bmatrix (where a > 0) has a negative eigenvalue if a is…
GATE 2023 ME Q32– A linear transformation maps a point (x, y) in the plane to the point (x, y) according to…
GATE 2022 ME (ME1) Q15– If A = bmatrix 10 & 2k + 5 \\ 3k - 3 & k + 5 bmatrix is a symmetric matrix, the value of k…
GATE 2022 ME (ME1) Q46– The system of linear equations in real (x, y) given by (x\ \ y)bmatrix 2 & 5 - 2 \\ & 1…