\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
We consider a system with three states \(e_1, e_2, e_3\) and a partially completed transition matrix:$$ \mathbf{M} = \begin{pmatrix} 0.2 & x & 0.5 \\ y & 0.8 & 0.1 \\ 0.4 & 0.4 & z \end{pmatrix} $$
  1. Determine the values of \(x, y,\) and \(z\) so that \(\mathbf{M}\) is a stochastic matrix.
  2. Once these values are found, what is the probability \(P(X_{n+1}=e_1 \mid X_n=e_2)\)?

Capture an image of your work. AI teacher feedback takes approximately 10 seconds.