Chapter 7.5, Problem 91E

To determine

To compute and interpret $P^2$ .

Answer to Problem 91E

  $P^2=\left[\begin{array}{ccc}0.4& 0.15& 0.15\\ 0.28& 0.53& 0.17\\ 0.32& 0.32& 0.68\end{array}\right]$

The matrix gives the proportions of the voting population that changed parties or remained loyal to their parties from the first election to the third..

Explanation of Solution

Given information:

The stochastic matrix is given

  

Each entry $P_{ij}(i\ne j)$ represents the proportion of the voting population that changes from party $i$ to party $j$ , and $p_{ii}$ represents the proportion that remains loyal to the party from one election to the next.

Concept used:

The product of matrix A and B exist only if the number of columns in matrix A is equal to the number of rows in matrix B .

Calculations:

Compute $P^2$ as shown:

  $\begin{array}{l}{P}^2=\left[\begin{array}{ccc}0.6& 0.1& 0.1\\ 0.2& 0.7& 0.1\\ 0.2& 0.2& 0.8\end{array}\right]\left[\begin{array}{ccc}0.6& 0.1& 0.1\\ 0.2& 0.7& 0.1\\ 0.2& 0.2& 0.8\end{array}\right]\\ =\left[\begin{array}{ccc}0.6\times 0.6+0.1\times 0.2+0.1\times 0.2& 0.6\times 0.1+0.1\times 0.7+0.1\times 0.2& 0.6\times 0.1+0.1\times 0.1+0.1\times 0.8\\ 0.2\times 0.6+0.7\times 0.2+0.1\times 0.2& 0.2\times 0.1+0.7\times 0.7+0.1\times 0.2& 0.2\times 0.1+0.7\times 0.1+0.1\times 0.8\\ 0.2\times 0.6+0.2\times 0.2+0.8\times 0.2& 0.2\times 0.1+0.2\times 0.7+0.8\times 0.2& 0.2\times 0.1+0.2\times 0.1+0.8\times 0.8\end{array}\right]\\ =\left[\begin{array}{ccc}0.4& 0.15& 0.15\\ 0.28& 0.53& 0.17\\ 0.32& 0.32& 0.68\end{array}\right]\end{array}$

Conclusion:

Thus, the matrixgives the proportions of the voting population that changed parties or remained loyal to their parties from the first election to the third.