Question

Let \(S = \left( {\begin{aligned}{*{20}{c}}0&1&0&0&0\\0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1\\0&0&0&0&0\end{aligned}} \right)\). Compute \({S^k}\) for \(k = {\bf{2}},...,{\bf{6}}\).

Short answer

The values are\({S^2} = \left( {\begin{aligned}{*{20}{c}}0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1\\0&0&0&0&0\\0&0&0&0&0\end{aligned}} \right)\),\({S^3} = \left( {\begin{aligned}{*{20}{c}}0&0&0&1&0\\0&0&0&0&1\\0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\end{aligned}} \right)\),\({S^4} = \left( {\begin{aligned}{*{20}{c}}0&0&0&0&1\\0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\end{aligned}} \right)\),\({S^5} = \left( {\begin{aligned}{*{20}{c}}0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\end{aligned}} \right)\), and\({S^6} = \left( {\begin{aligned}{*{20}{c}}0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\end{aligned}} \right)\).

Step-by-step solution

Create the matrix

Consider the matrix\(S = \left( {\begin{aligned}{*{20}{c}}0&1&0&0&0\\0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1\\0&0&0&0&0\end{aligned}} \right)\).

Write the above matrix in the MATLAB command.

\( > > {\rm{S}} = \left( {{\rm{0 1 0 0 0; 0 0 1 0 0; 0 0 0 1 0; 0 0 0 0 1; 0 0 0 0 0}}} \right){\rm{;}}\)

Obtain \({S^k}\) using \(k = {\bf{2}}\)

Put\(k = 2\)to obtain the matrix\({S^2}\)by using the MATLAB command shown below:

\( > > S*S\)

The output matrix is shown below:

\({S^2} = \left( {\begin{aligned}{*{20}{c}}0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1\\0&0&0&0&0\\0&0&0&0&0\end{aligned}} \right)\)

Thus, \({S^2} = \left( {\begin{aligned}{*{20}{c}}0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1\\0&0&0&0&0\\0&0&0&0&0\end{aligned}} \right)\).

Obtain \({S^k}\) using \(k = {\bf{3}}\)

Put\(k = 3\)to obtain the matrix\({S^3}\)by using the MATLAB command shown below:

\( > > S\^{\bf{3}}\)

The output matrix is shown below:

\({S^3} = \left( {\begin{aligned}{*{20}{c}}0&0&0&1&0\\0&0&0&0&1\\0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\end{aligned}} \right)\)

Thus, \({S^3} = \left( {\begin{aligned}{*{20}{c}}0&0&0&1&0\\0&0&0&0&1\\0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\end{aligned}} \right)\).

Obtain \({S^k}\) using \(k = {\bf{4}}\)

Put\(k = 4\)to obtain the matrix\({S^4}\)by using the MATLAB command shown below:

\( > > S\^4\)

The output matrix is shown below:

\({S^4} = \left( {\begin{aligned}{*{20}{c}}0&0&0&0&1\\0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\end{aligned}} \right)\)

Thus, \({S^4} = \left( {\begin{aligned}{*{20}{c}}0&0&0&0&1\\0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\end{aligned}} \right)\).

Obtain \({S^k}\) by using \(k = 5\)

Put\(k = 5\)to obtain the matrix\({S^5}\)by using the MATLAB command shown below:

\( > > S\^5\)

The output matrix is shown below:

\({S^5} = \left( {\begin{aligned}{*{20}{c}}0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\end{aligned}} \right)\)

Thus, \({S^5} = \left( {\begin{aligned}{*{20}{c}}0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\end{aligned}} \right)\).

Obtain \({S^k}\) using \(k = {\bf{6}}\)

Put\(k = 6\)to obtain the matrix\({S^6}\)by using the MATLAB command shown below:

\( > > S\^{\bf{6}}\)

The output matrix is shown below:

\({S^6} = \left( {\begin{aligned}{*{20}{c}}0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\end{aligned}} \right)\)

Thus, \({S^6} = \left( {\begin{aligned}{*{20}{c}}0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\end{aligned}} \right)\).