Question

If \(P\) is an orthogonal matrix of order \(3\times3\), \(A=\begin{bmatrix}1& 2& 2\\2& 1& 2\\2& 2& 1\end{bmatrix}\) and the eigenvalues of \(P^{T}AP\) are \(\alpha,\beta,\gamma\), then \(\alpha^2+\beta^2+\gamma^2=\)

• A. 3
• B. 27
• C. 15
• D. 12

Hint:

If \(A\) has eigenvalues \(\lambda_1,\lambda_2,\ldots,\lambda_n\), then \[ \boxed{\lambda_1^2+\lambda_2^2+\cdots+\lambda_n^2=\operatorname{tr}(A^2).} \] Also, orthogonally similar matrices have identical eigenvalues.

Answer 1

The Correct Option is B