Let (mathbf{Z}={0,1}) be the state space and [mathbf{P}(t)=left(begin{array}{cc} e^{-t} & 1-e^{-t} 1-e^{-t} & e^{-t} end{array} ight)]

Let \(\mathbf{Z}=\{0,1\}\) be the state space and

\[\mathbf{P}(t)=\left(\begin{array}{cc} e^{-t} & 1-e^{-t} \\ 1-e^{-t} & e^{-t} \end{array}\right)\]

the transition matrix of a continuous-time stochastic process \(\{X(t), t \geq 0\}\). Check whether \(\{X(t), t \geq 0\}\) is a homogeneous Markov chain.