Does relation S is transitive

Question

Let S be relation whose representing matrix $$ \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 1 \\ 1 & 0 & 1 \\ \end{bmatrix} $$ Does relation S is transitive?

I write down relation S: $$S = \{(1, 3), (2, 1), (2, 3), (3, 1), (3, 3)\}$$ I have:$$(2, 1), (1, 3)∈ S\quad\And\quad(2, 3)∈ S$$

$$(2, 3), (3, 1)∈ S\quad\And\quad(2, 1)∈ S$$ So it should be transitive, but the answer is not. Where did i miss?

Answer 1

We have $(1,3),(3,1) \in S$, but $(1,1)\not\in S$. You need to check all possible pairs.

Answer 2

You missed that $(1,3),(3,1)\in S$, but $(1,1)\notin S$.