Given

and the initial-state matrix S_{o} =

Find S_{1} for the given initial-state matrix S_{o} and explain what it represents.

**Solution:-**

Recall that S_{1} is the proportion of the population that is in each state after 1 trial. To calculate S_{1}, multiply the initial state matrix, S_{0}, by the transition matrix, P.

S_{1} = S_{0}P

=

Now use matrix multiplication to calculate S_{1}.

S_{1} =

=

Simplify each entry in the first state matrix to finish calculating S_{1}.

S_{1} =

=

=

Therefore S_{1} is .