Let $A=\{a,b,c\}$ and the relation given by the following matrix: $\mathcal M_r=\begin{pmatrix}1&0&1\\0&1&0\\1&1&0\end{pmatrix}$
The task is to find the smallest transitive relation $E$ such that $r\subseteq E$. Which is equivalent of saying find the transitive closure of $r$.
so what I know is that: $t(r)=\cup_{n\geq0}r^n, r^n=r\circ r\circ r\circ...\circ r, n$ times.
so $\mathcal M_{t(r)}=\sum_{n\in\mathbb{N}}\mathcal (M_r)^n$ and this gives me $$\mathcal M_{t(r)}=\begin{pmatrix}1&1&1\\0&1&0\\1&1&1\end{pmatrix}$$
Is there something wrong in my approach?