Then we take the 2nd row of the matrix, multiply it by the same vector and the sum becomes the 2nd element in the result vector. In the same way we can estimate all elements of the result vector. Each column specifies the fate of organisms in specific state. In the Leslie model, organisms' state is defined by age only. Leslie model is more complex because the sum of matrix elements in each column is not necessary equal to 1.
Matrix models are easy to iterate in time.