Question

To calculate the matrix \({R^{ - 1}}\), representing the inverse of the relation \(R\), whose matrix is \(R\).

Short answer

The matrix \({R^{ - 1}}\) is the transpose of the matrix \(R\).

Step-by-step solution

Given data

The matrix \(R\) of the relation \(R\).

Concept of Matrix relation

The matrix of the relation is given as:

\({R_{ij}} = 1 \Leftrightarrow (i,j) \in R\) …….(1)

\(R_{ij}^{ - 1} = 1 \Leftrightarrow {R_{ji}} = 1\) …….(2)

Calculation of the matrix \({R^{ - 1}}\)

From (1) and (2), the matrix \({R^{ - 1}}\) is obtained by interchange the rows and the columns of the matrix \(R\).

For example, if

\(R = \left( {\begin{array}{*{20}{l}}1&0&1\\1&0&1\\0&1&0\end{array}} \right)\), then \({R^{ - 1}} = \left( {\begin{array}{*{20}{l}}1&1&0\\0&0&1\\1&1&0\end{array}} \right)\).

Conclusion:

The matrix of the inverse relation is the transpose of the matrix of the relation \(R\).