Could the given matrix be the transition matrix of a regular Markov chain? \[ \left[\begin{array}{rr} 0.1 & 0.9 \\ 1 & 0 \end{array}\right] \]

Step 1 ：We are given the matrix \(\left[\begin{array}{rr} 0.1 & 0.9 \\ 1 & 0 \end{array}\right]\).

Step 2 ：We need to determine if this matrix could be the transition matrix of a regular Markov chain.

Step 3 ：A regular Markov chain is a Markov chain where it is possible to get from any state to any other state in a finite number of steps.

Step 4 ：In other words, a Markov chain is regular if its transition matrix to some power has all positive entries.

Step 5 ：The given matrix is a transition matrix of a Markov chain if the sum of each row is 1.

Step 6 ：To check if the given matrix is a transition matrix of a regular Markov chain, we need to check if the sum of each row is 1 and if there is a power of the matrix that has all positive entries.

Step 7 ：Checking the given matrix, we find that the sum of each row is indeed 1, so it is a transition matrix.

Step 8 ：Next, we need to check if there is a power of the matrix that has all positive entries.

Step 9 ：Upon checking, we find that there is indeed a power of the matrix that has all positive entries.

Step 10 ：Therefore, the given matrix could be the transition matrix of a regular Markov chain.

Step 11 ：\(\boxed{\text{Yes, the given matrix could be the transition matrix of a regular Markov chain.}}\)