Question #41104

Let $S$ be the set of all $3 \times 3$ matrices over $\mathbb{R}$ with entries from $\{0, 1\}$. We define three subsets of $S$ as follows: $X = \{ A \in S \mid \det(A) = 0 \text{ and } \det(R_{12}(A)) \neq 0 \}$ $Y = \{ A \in S \mid \det(A) \neq 0 \text{ and } \det(R_{12}(A)) = \det(A) \}$ $W = \{ A \in S \mid A^T = -A \text{ and } A \neq O \}$ where $R_{12}(A)$ denotes the matrix obtained by swapping the first and second rows of $A$, and $O$ is the $3 \times 3$ null matrix. Let $Z = X \cup Y \cup W$. Let $n$ be the number of reflexive relations on $Z$, and $m$ be the number of symmetric relations on $Z$. Find the value of $n + m$.