GATE 2025 MA – Question 62
M is the symmetric matrix of Q=2x₁²+2x₂²+3x₃²+4x₁x₂+2x₁x₃+2x₂x₃. For any nonzero Y∈ℝ³, $a_n=Y^T(M+I)^{n+1}Y/[Y^T(M+I)^nY]$. Find lim a_n (integer).
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Correct answer: 6 to 6
Explanation
$M=\begin{pmatrix}2&2&1\\2&2&1\\1&1&3\end{pmatrix}$ has eigenvalues 0,2,5, with eigenvectors (1,−1,0),(1,1,−2),(1,1,1). Thus M+I has eigenvalues 1,3,6. In an orthonormal eigenbasis, a_n is a ratio of sums of nonnegative squared projections times eigenvalue powers; its limit is the largest eigenvalue with nonzero projection. The official answer **6** assumes Y has nonzero projection on (1,1,1). For Y=(1,−1,0) the limit is 1, and for Y=(1,1,−2) it is 3. The arbitrary-nonzero-vector wording omits this assumption; official grading remains 6.