GATE 2026 PI – Question 38
Match the matrices P,Q,R,S to properties 1: singular; 2: triangular; 3: symmetric; 4: trace-free. $$P=\begin{pmatrix}-30&17&5\\17&0&-12\\5&-12&4\end{pmatrix},\quad Q=\begin{pmatrix}0&1&9\\7&0&2\\12&3&0\end{pmatrix}.$$ $$R=\begin{pmatrix}2/3&1/3&2/3\\0&2/3&-1/3\\0&4/3&-2/3\end{pmatrix},\quad S=\begin{pmatrix}1&2&3\\0&1&4\\0&0&1\end{pmatrix}.$$
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Correct answer: (C) P–3; Q–4; R–1; S–2
Explanation
P equals its transpose, so it is symmetric. Q has zero diagonal sum and is trace-free. In R the third row is twice the second, so its determinant is zero and it is singular. S has all entries below its diagonal zero and is upper triangular. Thus **P–3, Q–4, R–1, S–2: C**.