Is the matricial representation of the inverse relation the transpose of the representation of the original relation?

Question

here we are studying relations on AxA with A a finite set. We represent the relation on a matrix in this way:

$M\left ( R \right )_{ij}=1 \; if (i,j) \in R; and M\left ( R \right )_{ij}=0 \;\;otherwise $

It seems that the representation of the inverse relation $$ R^{-1} = \{\langle x,y\rangle\, |\,\langle y,x\rangle \in R\ \} $$ should be the transpose of the matrix of the original relation.

As we could not find it in any book or link, we post the question ( sorry about a bad english )

Answer 1

Yes, you are right: the matrix of the inverse relation $R^{-1}$ is indeed $R^T$.