Question:

Let $T=\{\alpha_1,\alpha_2,\beta\}$ be the 3 distinct roots of $x^2+x-1=0$ (note: a quadratic has at most 2 roots; here the set $T$ includes specific values). For a $3\times 3$ matrix $M=(a_{ij})$, let $R_i=a_{i1}+a_{i2}+a_{i3}$ and $C_j=a_{1j}+a_{2j}+a_{3j}$. Match entries in List-I with List-II: P) Number of $M$ with all entries in $T$ such that $R_i=C_j=0$ for all $i,j$ Q) Number of symmetric $M$ with all entries in $T$ such that $C_j=0$ for all $j$ R) Skew-symmetric $M$ with $a_{ij}\in T$ for $i>j$ — number of solutions to $M(x,y,z)^T=(-a,a,0)^T$ S) $M$ with all entries in $T$, $R_i=0$ for all $i$ — absolute value of $\det(M)$ List-II: 1)1, 2)12, 3)∞, 4)6, 5)0