&\c{k+1}{k-1}&\c{k+2}{k-1}&\cdots&\c{n}{k-1}&\c{k}{k-1} \\  &   & &  & 0 & \c{k+2}{k+1}&\cdots&\c{n}{k+1}& 0\\
  &   & &  & & 0 & \vdots& \vdots& \vdots\\
  &   & &  &  & &\ddots & \c{n}{n-1}& 0\\
  &   & &  & & & &0&0\\  &   & &  & & & &&0
\end{array}\right).$$

Note that $\Lambda'$ is obtained from $\Lambda$ by interchanging the $k$-th and the $n$-th rows and the $k$-th and the $n$-th columns. So $\Lambda'$ is conjugate to $\Lambda$, and again, by [@ch-ma-su08] and [@MP], we can see that their corresponding Bott manifolds are diffeomorphic. However, we give an elementary and direct indication of proof here for reader’s convenience.

This is an easy consequence of Lemma \[lemma:2-1\]. The only thing to consider is that when exchanging the columns we need to take care of the effect of the indices of $x_j$’s. Here we only give an idea of the proof with an example. The proof of the general case is quite similar. Here we consider $B_4$ with the following associated matrix $$A=
\left(
\begin{array}{c c c c}
0 & a & b & c \\
 & 0 & 0 & 0 \\
 &  & 0 & d \\
 &  &  & 0
\end{array}
\right).$$ Then $H^\ast(B_4)\cong \mathbb 