91Ó°ÊÓ

For any transition matrix, the sum of the entries of the columns is equal to 1 .

Let there be any 3x3 transition matrix as $A=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]$.

Then,

$$\begin{gathered} \left[111\right]A=\left[111\right]\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right] \\ =\left[\begin{array}{ccc}{a}_{11}+a_{21}+a_{31}& a_{12}+a_{22}+a_{32}& a_{13}+a_{23}+a_{33}\end{array}\right] \\ =\left[111\right] \end{gathered}$$

Hence, the matrix product is: $\left[111\right]$.

The given row vector is$\stackrel{→}{e}=\frac{\left[1,1,..,1\right]}{n1\text{'}s}$ .

Let there A be any nxn matrix A with non-zero elements. Then, we have:

$\stackrel{→}{e}A=\left[\underset{i=1}{\overset{n}{∑a_{i1,}}}\underset{i=1}{\overset{n}{∑a_{i2,}}}⋯\underset{i=1}{\overset{n}{∑a_{im,}}}\right]$

Let, A be the transition matrix. Then, sum of its columns will be 1.

So,

$$\begin{gathered} \stackrel{→}{e}A=\left[\underset{i=1}{\overset{n}{∑a_{i1,}}}\underset{i=1}{\overset{n}{∑a_{i2,}}}⋯\underset{i=1}{\overset{n}{∑a_{im,}}}\right] \\ =\left[1,1,...1\right] \\ =\stackrel{→}{e} \end{gathered}$$

Again, there be any nxn matrix A with non-zero elements. Then, we have:

$\stackrel{→}{e}A=\left[\underset{i=1}{\overset{n}{∑a_{i1,}}}\underset{i=1}{\overset{n}{∑a_{i2,}}}⋯\underset{i=1}{\overset{n}{∑a_{im,}}}\right]$

Let us assume, $\stackrel{→}{e}A=\stackrel{→}{e}$. Then, sum of its columns will be 1.

So,

$$\begin{gathered} \left[\underset{i=1}{\overset{n}{∑a_{i1,}}}\underset{i=1}{\overset{n}{∑a_{i2,}}}⋯\underset{i=1}{\overset{n}{∑a_{in,}}}\right]=\stackrel{→}{e}A \\ =\stackrel{→}{e} \\ =\left[1,1,....1\right] \end{gathered}$$

Hence, A is a transition matrix.