(a) since \(p+1-p=1\) and \(q+1-q=1,\) the first matrix \(\mathbf{A}\) is
stochastic. since \(\frac{1}{2}+\frac{1}{4}+\frac{1}{4}=1,
\frac{1}{3}+\frac{1}{3}+\frac{1}{3}=1\), and
\(\frac{1}{6}+\frac{1}{3}+\frac{1}{2}=1,\) the second matrix \(\mathbf{A}\) is
stochastic.
(b) The matrix from part (a) is shown with its eigenvalues and corresponding
eigenvectors. \(\left(\begin{array}{ccc}\frac{1}{2} & \frac{1}{4} & \frac{1}{4}
\\\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ \frac{1}{6} & \frac{1}{3} &
\frac{1}{2}\end{array}\right) ; \quad\) eigenvalues: \(1,
\frac{1}{6}-\frac{1}{12} \sqrt{2}, \frac{1}{6}+\frac{1}{12} \sqrt{2};\)
eigenvectors: \((1,1,1),\left(-\frac{3(-1+\sqrt{2})}{-6+\sqrt{2}},
\frac{2(2+\sqrt{2})}{-6+\sqrt{2}},
1\right),\left(-\frac{3(1+\sqrt{2})}{6+\sqrt{2}},
\frac{2(-2+\sqrt{2})}{6+\sqrt{2}}, 1\right)\) Further examples indicate that 1
is always an eigenvalue with corresponding eigenvector \((1,1,1) .\) To prove
this, let \(\mathbf{A}\) be a stochastic matrix and \(\mathbf{K}=(1,1,1) .\) Then
\(\mathbf{A} \mathbf{K}=\left(\begin{array}{ccc}a_{11} & \cdots & a_{1 n} \\\
\vdots & & \vdots \\ a_{n 1} & \cdots & a_{n
n}\end{array}\right)\left(\begin{array}{c}1 \\ \vdots \\\
1\end{array}\right)=\left(\begin{array}{c}a_{11}+\cdots+a_{1 n} \\ \vdots \\\
a_{n 1}+\cdots+a_{n n}\end{array}\right)=\left(\begin{array}{c}1 \\ \vdots
\\\ 1\end{array}\right)=1 \mathbf{K},\) and 1 is an eigenvalue of \(\mathbf{A}\)
with corresponding eigenvector (1,1,1).
(c) For the \(3 \times 3\) matrix in part (a) we have
\(\mathbf{A}^{2}=\left(\begin{array}{ccc}\frac{3}{8} & \frac{7}{24} &
\frac{1}{3} \\ \frac{1}{3} & \frac{11}{36} & \frac{13}{36} \\ \frac{5}{18} &
\frac{23}{72} & \frac{29}{72}\end{array}\right), \quad
\mathbf{A}^{3}=\left(\begin{array}{ccc}\frac{49}{144} & \frac{29}{96} &
\frac{103}{288} \\ \frac{71}{216} & \frac{11}{36} & \frac{79}{216} \\\
\frac{5}{16} & \frac{67}{216} & \frac{163}{432}\end{array}\right).\) These
powers of A are also stochastic matrices. To prove that this is true in
general for \(2 \times 2\) matrices, we prove the more general theorem that any
product of \(2 \times 2\) stochastic matrices is stochastic. Let
\(\mathbf{A}=\left(\begin{array}{ll}a_{11} & a_{12} \\ a_{21} &
a_{22}\end{array}\right) \quad\) and \(\quad
\mathbf{B}=\left(\begin{array}{ll}b_{11} & b_{12} \\ b_{21} &
b_{22}\end{array}\right)\) be stochastic matrices. Then \(\mathbf{A
B}=\left(\begin{array}{ll}a_{11} b_{11}+a_{12} b_{21} & a_{11} b_{12}+a_{12}
b_{22} \\ a_{21} b_{11}+a_{22} b_{21} & a_{21} b_{12}+a_{22}
b_{22}\end{array}\right).\) The sums of the rows are \(\begin{aligned} a_{11}
b_{11}+a_{12} b_{21}+a_{11} b_{12}+a_{12} b_{22}
&=a_{11}\left(b_{11}+b_{12}\right)+a_{12}\left(b_{21}+b_{22}\right) \\\
&=a_{11}(1)+a_{12}(1)=a_{11}+a_{12}=1 \\ a_{21} b_{11}+a_{22} b_{21}+a_{21}
b_{12}+a_{22} b_{22}
&=a_{21}\left(b_{11}+b_{12}\right)+a_{22}\left(b_{21}+b_{22}\right) \\\
&=a_{21}(1)+a_{22}(1)=a_{21}+a_{22}=1. \end{aligned}\) Thus, the product matrix
\(\mathbf{A B}\) is stochastic. It follows that any power of a \(2 \times 2\)
matrix is stochastic. The proof in the case of an \(n \times n\) matrix is very
similar.
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