Let P $=$

$$

$$

$1$

$2$

$1$

$3$

$1$

$3$

$01$

$3$

$2$

$3$

$1$

$2$

$1$

$30$

$$

$$ be the transition matrix for a Markov chain with three states and $$x $(0)=$

$$

$$

$120$

$180$

$90$

$$

$$ as

the initial state vector for the population. Compute $$x $(1),$x $(2),$ and $$x $(3)$ exactly and $$x $(10)$ rounded to $10$

digits.