Regular Transition Matrix 
A transition matrix is regular if some power of the matrix contains
all positive entries. [1]


Its powers have all positive entries...

Why isn't this matrix a Regular Transition Matrix?
Reference:
[1] http://fazekas-andras-istvan.hu/9_11_markov_lancok/DFAI_MARKOV_CHAINS_02.pdf

It can be shown that (either by theoretical derivation or using Matlab) $B$ has the Jordan canonical form decomposition as follows:
$$B = PJP^{-1},$$
where 
\begin{align*}
P = \begin{bmatrix}
1 & 0 & 1 \\
0 & 1 & 0 \\
0 & 0 & 1 
\end{bmatrix},
\end{align*}
\begin{align*}
J = \begin{bmatrix}
0.5 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix}.
\end{align*}
Hence for each natural number $n$, we have:
\begin{align*}
B^n = & (PJP^{-1})^n = PJ^nP^{-1} \\
= & \begin{bmatrix}
1 & 0 & 1 \\
0 & 1 & 0 \\
0 & 0 & 1 
\end{bmatrix}
\begin{bmatrix}
0.5^n & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
1 & 0 & -1 \\
0 & 1 & 0 \\
0 & 0 & 1 
\end{bmatrix} \\
= &
\begin{bmatrix}
0.5^n & 0 & 1 - 0.5^n \\
0 & 1 & 0 \\
0 & 0 & 1 
\end{bmatrix}.
\end{align*}
It thus can be seen that not every entry of $B^n$ is positive, hence $B$ is not a regular transition matrix.
