How to properly solve this Hidden Markov Model problem?

Question

I got a an exercise problem which should be seen as a HMM scenario and argument some statements. However I'm quite confused about how to properly solve and argument my solutions.

Problem tells:

Imagine you want to determine the annual temperature centuries of years ago, when of course there wasn't any thermometer or records. So, nature as an evidence is a worth to try resource, we may achieve it by watching at tree's inside rings. There's reliable evidence suggesting that there's a relation among the rings inside trees and temperature. There will be 2 different temperature states, WARM (W) and COLD (C) and three discretized tree rings sizes: SMALL (S), MEDIUM (M) and LARGE (L). Some researchers have provided two matrixes:

$\begin{bmatrix}.7 & .3\\.4 & .6\end{bmatrix}$

As transition matrix, so the probability of remaining in COLD state if COLD is present is $.6$ and the probability of passing from COLD to WARM is $.7$.

Also, a second matrix with the relation among the ring's size and the temperature over the year:

$\begin{bmatrix}.1 & .4 & .5\\.7 & .2 & .1\end{bmatrix}$

So, problem asks what I should do to calculate the chance of a sequence (for example):

SSSMMLLL

To happen. I considered multiplying the 4 distinct cases and so generate a Markov matrix with transitions among trees ring sizes. However I never got a matrix whose rows sum 1 as it should.

How could I solve this?