Question

Describe in words what happens when you compute \({A^{\bf{5}}}\), \({A^{{\bf{10}}}}\), \({A^{{\bf{20}}}}\), and \({A^{{\bf{30}}}}\) for \(A = \left( {\begin{aligned}{*{20}{c}}{1/6}&{1/2}&{1/3}\\{1/2}&{1/4}&{1/4}\\{1/3}&{1/4}&{5/12}\end{aligned}} \right)\).

Short answer

As the power increases, the matrix becomes more like \(\left( {\begin{aligned}{*{20}{c}}{1/3}&{1/3}&{1/3}\\{1/3}&{1/3}&{1/3}\\{1/3}&{1/3}&{1/3}\end{aligned}} \right)\).

Step-by-step solution

Create the matrix

Consider the matrix\(A = \left( {\begin{aligned}{*{20}{c}}{1/6}&{1/2}&{1/3}\\{1/2}&{1/4}&{1/4}\\{1/3}&{1/4}&{5/12}\end{aligned}} \right)\).

Write the above matrix in MATLAB command.

\( > > {\rm{A}} = \left( {{\rm{1/6 1/2 1/3; 1/2 1/4 1/4; 1/3 1/4 5/12}}} \right){\rm{;}}\)

Obtain the matrix by using the MATLAB command

Obtain the matrix\({A^5}\)by using the MATLAB command shown below:

\( > > A\^{\bf{5}}\)

The output matrix is shown below:

\({A^5} = \left( {\begin{aligned}{*{20}{c}}{0.331822273662551}&{0.334587191368025}&{0.333590534979424}\\{0.334587191358025}&{0.332320601851852}&{0.333092206790123}\\{0.333590534979424}&{0.333092206790123}&{0.333317258230453}\end{aligned}} \right)\)

Thus, \({A^5} = \left( {\begin{aligned}{*{20}{c}}{0.331822273662551}&{0.334587191368025}&{0.333590534979424}\\{0.334587191358025}&{0.332320601851852}&{0.333092206790123}\\{0.333590534979424}&{0.333092206790123}&{0.333317258230453}\end{aligned}} \right)\).

Obtain the matrix by using the MATLAB command

Obtain the matrix\({A^{10}}\)by using the MATLAB command shown below:

\( > > A\^10\)

The output matrix is shown below:

\({A^{10}} = \left( {\begin{aligned}{*{20}{c}}{0.333337254997295}&{0.333330106839401}&{0.333332638213305}\\{0.333330106839401}&{0.333335989260343}&{0.333333903900257}\\{0.333332638213305}&{0.333333903900257}&{0.333333457886439}\end{aligned}} \right)\)

Thus, \({A^{10}} = \left( {\begin{aligned}{*{20}{c}}{0.333337254997295}&{0.333330106839401}&{0.333332638213305}\\{0.333330106839401}&{0.333335989260343}&{0.333333903900257}\\{0.333332638213305}&{0.333333903900257}&{0.333333457886439}\end{aligned}} \right)\).

Obtain the matrix by using the MATLAB command

Obtain the matrix\({A^{20}}\)by using the MATLAB command shown below:

\( > > A\^{\bf{20}}\)

The output matrix is shown below:

\({A^{20}} = \left( {\begin{aligned}{*{20}{c}}{0.33333}&{0.33333}&{0.33333}\\{0.33333}&{0.33333}&{0.33333}\\{0.33333}&{0.33333}&{0.33333}\end{aligned}} \right)\)

Thus, \({A^{20}} = \left( {\begin{aligned}{*{20}{c}}{0.33333}&{0.33333}&{0.33333}\\{0.33333}&{0.33333}&{0.33333}\\{0.33333}&{0.33333}&{0.33333}\end{aligned}} \right)\).

Obtain the matrix by using the MATLAB command

Obtain the matrix\({A^{30}}\)by using the MATLAB command shown below:

\( > > A\^{\bf{30}}\)

The output matrix is shown below:

\({A^{30}} \approx \left( {\begin{aligned}{*{20}{c}}{0.33333}&{0.33333}&{0.33333}\\{0.33333}&{0.33333}&{0.33333}\\{0.33333}&{0.33333}&{0.33333}\end{aligned}} \right)\)

Thus,\({A^{30}} = \left( {\begin{aligned}{*{20}{c}}{0.33333}&{0.33333}&{0.33333}\\{0.33333}&{0.33333}&{0.33333}\\{0.33333}&{0.33333}&{0.33333}\end{aligned}} \right)\).

It is observed that as the power increases, the matrix becomes more like \(\left( {\begin{aligned}{*{20}{c}}{0.3333}&{0.3333}&{0.3333}\\{0.3333}&{0.3333}&{0.

3333}\\{0.3333}&{0.3333}&{0.3333}\end{aligned}} \right)\)or \(\left( {\begin{aligned}{*{20}{c}}{1/3}&{1/3}&{1/3}\\{1/3}&{1/3}&{1/3}\\{1/3}&{1/3}&{1/3}\end{aligned}} \right)\).

All the entries in\({A^{20}}\)and\({A^{30}}\)approach to 0.3333.