Question

To find

a) The matrix representing \({R^2}\).

b) The matrix representing \({R^3}\).

c) The matrix representing \({R^4}\).

Short answer

a) The matrix is \({M_{{R^2}}} = \left( {\begin{array}{*{20}{l}}0&0&1\\1&1&0\\0&1&1\end{array}} \right)\).

b) The matrix is \({M_{{R^3}}} = \left( {\begin{array}{*{20}{l}}1&1&0\\0&1&1\\1&1&1\end{array}} \right)\).

c) The matrix is \({M_{{R^4}}} = \left( {\begin{array}{*{20}{l}}0&1&1\\1&1&1\\1&1&1\end{array}} \right)\).

Step-by-step solution

Given data

The matrix \({M_R}\) of the relation \(R\).

Concept of Matrix 

The ordered pair\((i,j)\)belongs to the relation if and only if the\({(i,j)^{th}}\)entry in the matrix is\(1\).

Calculation for the matrix \({R^2}\) 

a)

From matrix formula, we have:

\({M_{{R^2}}} = {M_R}{M_R} = \left( {\begin{array}{*{20}{l}}0&1&0\\0&0&1\\1&1&0\end{array}} \right)\left( {\begin{array}{*{20}{l}}0&1&0\\0&0&1\\1&1&0\end{array}} \right) = \left( {\begin{array}{*{20}{l}}0&0&1\\1&1&0\\0&1&1\end{array}} \right)\)

Calculation for matrix \({R^3}\) 

b)

From formula and a), we have:

\({M_{{R^3}}} = {M_R}{M_{{R^2}}} = \left( {\begin{array}{*{20}{l}}0&1&0\\0&0&1\\1&1&0\end{array}} \right)\left( {\begin{array}{*{20}{l}}0&0&1\\1&1&0\\0&1&1\end{array}} \right) = \left( {\begin{array}{*{20}{l}}1&1&0\\0&1&1\\1&1&1\end{array}} \right)\)

Calculation for matrix \({R^4}\) 

c)

From formula and part b), we have:

\({M_{{R^4}}} = {M_R}{M_{{R^3}}} = \left( {\begin{array}{*{20}{l}}0&1&0\\0&0&1\\1&1&0\end{array}} \right)\left( {\begin{array}{*{20}{l}}1&1&0\\0&1&1\\1&1&1\end{array}} \right) = \left( {\begin{array}{*{20}{l}}0&1&1\\1&1&1\\1&1&1\end{array}} \right)\)