Find transitive closure of the relation, given its matrix

Question

Find transitive closure of relation $R$ described by the matrix $M_R$:

$$M_R = \begin{bmatrix}1 & 0 &0 \\0 & 1 & 1 \\1 & 0 & 1 \end{bmatrix}$$

I tried doing it like this $M_R \cdot M_R \cdot M_R$ but could not get the answer.

Answer 1

The relation indicated by this matrix can be described as $$ (x,x) \quad x = 1,2,3\\ (2,3)\\ (3,1) $$ In order for the relation to be transitively closed, you need to add $(2,1)$. So, the matrix we want is $$ \pmatrix{ 1&0&0\\ 1&1&1\\ 1&0&1 } $$