Consider the given matrices, \[M_{R1}=\begin{bmatrix}0&1&0\\1&1&1\\1&0&0\end{bmatrix}\ and\ M_{R2}=\begin{bmatrix}0&1&0\\0&1&1\\1&1&1\end{bmatrix}\]

Note that, the intersection \(\displaystyle{a}\cap{B}\) is all elements that are both in A and in B.

The matrix corresponding to the intersection of two relations is the meet of the matrices representing each of the relations.

\(\displaystyle{M}_{{{R}_{{1}}\cap\ {R}_{{2}}}}={M}_{{{R}_{{1}}}}\cap{M}_{{{R}_{{2}}}}\)

\[=\begin{bmatrix}0\wedge0&1\wedge1&0\wedge0\\1\wedge0&1\wedge1&1\wedge1\\1\wedge1&0\wedge1&0\wedge1\end{bmatrix}\]

\[=\begin{bmatrix}0&1&1\\0&1&1\\1&0&0\end{bmatrix}\]

Thus, the matrix that represents \[R_1\cap R_2\ is\ =\begin{bmatrix}0&1&1\\0&1&1\\1&0&0\end{bmatrix}\]