Find the zero-one matrix of the transitive closure of the relation given by the matrix $A$:

Given matrix is

$A=\left[\begin{array}{ccc}1 & 0 & 1\\0 & 1 & 0\\1 & 1 & 0\end{array}\right]$

From the matrix, the relation contains:

$1 \to 1,\ 1 \to 3,\ 2 \to 2,\ 3 \to 1,\ 3 \to 2$

Now, since $1 \to 3$ and $3 \to 2$, therefore $1 \to 2$ must be added.

Also, since $3 \to 1$ and $1 \to 3$, therefore $3 \to 3$ must be added.

So, the transitive closure matrix is

$\left[\begin{array}{ccc}1 & 1 & 1\\0 & 1 & 0\\1 & 1 & 1\end{array}\right]$

Hence, the correct answer is Option 1.