Rows and columns of the one-step transition probability matrix

Question

I am currently studying the textbook Introduction to Modeling and Analysis of Stochastic Systems, Second Edition, by V. G. Kulkarni. In a section on discrete-time Markov chains, the author introduces the one-step transition probability matrix as follows:

$$P = \begin{bmatrix} p_{1, 1} & p_{1, 2} & p_{1, 3} & \dots & p_{1, N} \\ p_{2, 1} & p_{2, 2} & p_{2, 3} & \dots & p_{2, N} \\ p_{3, 1} & p_{3, 2} & p_{3, 3} & \dots & p_{3, N} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ p_{N, 1} & p_{N, 2} & p_{N, 3} & \dots & p_{N, N} \end{bmatrix} \tag{2.4}$$

The matrix $P$ in the equation above is called the one-step transition probability matrix, or transition matrix for short, of the DTMC. Note that the rows correspond to the starting state and the columns correspond to the ending state of a transition.

I'm not sure what the author means by the following:

... the rows correspond to the starting state and the columns correspond to the ending state of a transition.

I would greatly appreciate it if people would please take the time to clarify this.

Answer 1

They're just saying that the probability of ending in state $j$, given that you start in state $i$ is the element in the $i$th row and $j$th column of the matrix. For example, if you start in state $3$, the probability of transitioning to state $7$ is the element in the 3rd row, and 7th column of the matrix: $p_{37}$.