JEE Advanced 2024 Mathematics question on Matrices and Determinants

Let $\alpha$ and $\beta$ be the distinct roots of the equation $x^2 + x - 1 = 0$. Consider the set $T = \{1, \alpha, \beta\}$. For a $3 \times 3$ matrix $M = (a_{ij})_{3\times 3}$, define $R_i = a_{i1} + a_{i2} + a_{i3}$ and $C_j = a_{1j} + a_{2j} + a_{3j}$ for $i = 1,2,3$ and $j = 1,2,3$.

Match each entry in List-I to the correct entry in List-II.
List-I
(P) The number of matrices $M = (a_{ij})_{3\times 3}$ with all entries in $T$ such that $R_i = C_j = 0$ for all $i,j$, is
(Q) The number of symmetric matrices $M = (a_{ij})_{3\times 3}$ with all entries in $T$ such that $C_j = 0$ for all $j$, is
(R) Let $M = (a_{ij})_{3\times 3}$ be a skew symmetric matrix such that $a_{ij} = 0$ for $i > j$. Then the number of elements in the set $\left\{ \begin{pmatrix} x \\ y \\ z \end{pmatrix} : x,y,z \in \mathbb{R}, M \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} a_{12} \\ 0 \\ -a_{23} \end{pmatrix} \right\}$ is
(S) Let $M = (a_{ij})_{3\times 3}$ be a matrix with all entries in $T$ such that $R_i = 0$ for all $i$. Then the absolute value of the determinant of $M$ is
List-II
(1) 1
(2) 12
(3) infinite
(4) 6
(5) 0