chapter 07 of the book "Introduction to Probability, 2nd Edition" introduces this figure
the red rectangle is added by myself.
which take i = 1 for this formula
$\sum_{j=1}^m p_{ij} = 1, \quad \text{for all i}$
and gets
$\sum_{j=1}^m p_{1j} = 1$
the interpretation of which could be, current is at state 1, the probability that the next state is equal to j (1, 2, ..., m)
p11 represents from state 1 (time instant_1) to state 1 (time instant_2)
p12 represents from state 1 (time instant_1) to state 2 (time instant_2)
...
p1m represents from state 1 (time instant_1) to state m (time instant_2)
the sum of all the above p11+p12+...+p1m = 1
the question is how about the column one?
I guess p11+p21+...+pm1 is also equal to 1, is it?
Keep reading, you'll find this example in the book "Introduction to Probability, 2nd Edition"
"Alice is taking a probability class and in each week, she can be either up-to-date or she may have fallen behind. If she is up-to-date in a given week, the probability that she will be up-to-date (or behind) in the next week is 0.8 (or 0.2, respectively). If she is behind in the given week, the probability that she will be up-to-date (or behind) in the next week is 0.6 (or 0.4, respectively). We assume that these probabilities do not depend on whether she was up-to-date or behind in previous weeks, so the problem has the typical Markov chain character (the future depends on the past only through the present)..."
"... let us introduce states 1 and 2, and identify them with being up-to-date and behind, respectively..." then the transition probability Matrix is
$$ \begin{bmatrix} 0.8 & 0.2 \\ 0.6 & 0.4 \\ \end{bmatrix} $$
The sum of each row is 1, whereas the sum of column is not.