Question

(M) Certain dynamical systems can be studied by examining powers of a matrix, such as those below. Determine what happens to \({A^k}\) and \({B^k}\) as \(k\) increases (for example, try \(k = 2,...,16\)). Try to identify what is special about \(A\) and \(B\). Investigate large powers of other matrices of this type, and make a conjecture about such matrices.

\(A = \left( {\begin{aligned}{*{20}{c}}{.4}&{.2}&{.3}\\{.3}&{.6}&{.3}\\{.3}&{.2}&{.4}\end{aligned}} \right),B = \left( {\begin{aligned

Short answer

As \(k\) increases, the matrices \({{\mathop{\rm A}\nolimits} ^k}\) and \({{\mathop{\rm B}\nolimits} ^k}\) become \({{\mathop{\rm A}\nolimits} ^k} = \left( {\begin{aligned}{*{20}{c}}{\frac{2}{7}}&{\frac{2}{7}}&{\frac{2}{7}}\\{\frac{3}{7}}&{\frac{3}{7}}&{\frac{3}{7}}\\{\frac{2}{7}}&{\frac{2}{7}}&{\frac{2}{7}}\end{aligned}} \right)\), \({{\mathop{\rm B}\nolimits} ^k} = \left( {\begin{aligned}{*{20}{c}}{\frac{{18}}{{89}}}&{\frac{{18}}{{89}}}&{\frac{{18}}{{89}}}\\{\frac{{33}}{{89}}}&{\frac{{33}}{{89}}}&{\frac{{33}}{{89}}}\\{\frac{{38}}{{89}}}&{\frac{{38}}{{89}}}&{\frac{{38}}{{89}}}\end{aligned}} \right)\).

Step-by-step solution

Determine what happens to \({A^k}\) as k increases

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

Use the MATLAB code to compute the matrices \({A^2},{A^4},{A^8}\) as shown below:

\(\begin{aligned}{l} > > {\mathop{\rm A}\nolimits} = \left( {.4\,\,\,.2\,\,\,.3;\,\,.3\,\,\,.6\,\,\,.3;\,\,.3\,\,\,.2\,\,\,.4} \right)\\ > > {\mathop{\rm A}\nolimits} \^2\end{aligned}\)

\({{\mathop{\rm A}\nolimits} ^2} = \left( {\begin{aligned}{*{20}{c}}{.31}&{.26}&{.30}\\{.39}&{.48}&{.39}\\{.30}&{.26}&{.31}\end{aligned}} \right)\)

\(\begin{aligned}{l} > > {\mathop{\rm A}\nolimits} \^2 = \left( {.31\,\,\,.26\,\,\,.30;\,\,.39\,\,\,.48\,\,\,.39;\,\,.30\,\,\,.26\,\,\,.31} \right)\\ > > {\mathop{\rm A}\nolimits} \^4 = {\mathop{\rm A}\nolimits} \^2 * {\mathop{\rm A}\nolimits} \^2\end{aligned}\)

\({{\mathop{\rm A}\nolimits} ^4} = \left( {\begin{aligned}{*{20}{c}}{.2875}&{.2834}&{.2874}\\{.4251}&{.4332}&{.4251}\\{.2874}&{.2834}&{.2875}\end{aligned}} \right)\)

\(\begin{aligned}{l} > > {\mathop{\rm A}\nolimits} \^4 = \left( {.2857\,\,.2857\,\,\,.2857;\,\,.4285\,\,.4286\,\,\,.4285;\,\,.2857\,\,\,.2857\,\,\,.2857} \right)\\ > > {\mathop{\rm A}\nolimits} \^8 = {\mathop{\rm A}\nolimits} \^4 * {\mathop{\rm A}\nolimits} \^4\end{aligned}\)

\({{\mathop{\rm A}\nolimits} ^8} = \left( {\begin{aligned}{*{20}{c}}{.2857}&{.2857}&{.2857}\\{.4285}&{.4286}&{.4285}\\{.2857}&{.2857}&{.2857}\end{aligned}} \right)\)

As \(k\)increases, the four decimal places become

\({{\mathop{\rm A}\nolimits} ^k} \to \left( {\begin{aligned}{*{20}{c}}{.2857}&{.2857}&{.2857}\\{.4286}&{.4286}&{.4286}\\{.2857}&{.2857}&{.2857}\end{aligned}} \right)\).

The rational format of the matrix \({A^k}\) is shown below:

\({{\mathop{\rm A}\nolimits} ^k} \to \left( {\begin{aligned}{*{20}{c}}{\frac{2}{7}}&{\frac{2}{7}}&{\frac{2}{7}}\\{\frac{3}{7}}&{\frac{3}{7}}&{\frac{3}{7}}\\{\frac{2}{7}}&{\frac{2}{7}}&{\frac{2}{7}}\end{aligned}} \right)\)

Determine what happens to \({B^k}\) as k increases

Consider the matrix.

Use the MATLAB code to compute the matrices as shown below: