I'm trying to follow an argument in Lück's "Algebraische Topologie: Homologie und Mannigfaltigkeiten" (to which there apparently doesn't exist an english translation). The aim is to check homotopy invariance of cellular homology by constructing a chain homotopy.

Let me sketch the argument. Let $h\colon (X,A)\times [0,1]\to (Y,B)$ be a cellular homotopy from $f_0$ to $f_1$. Using the CW-structure on $[0,1]$ with the two 0-cells {0}, {1} and one 1-cell, we identify $$ C_n((X,A)\times [0,1])=C_n(X,A)\oplus C_n(X,A)\oplus C_{n-1}(X,A). $$ Then $C_n(h)$ is of the form $C_n(f_0)\oplus C_n(f_1)\oplus u_{n-1}$, where $u_{n-1}$ is some map $C_{n-1}(X,A)\to C_n(Y,B)$.

Now we would like to compute the $n$th differential of $C_*((X,A)\times [0,1])$ under the above identification. Since it is a map from $C_n(X,A)\oplus C_n(X,A)\oplus C_{n-1}(X,A)$ to $C_{n-1}(X,A)\oplus C_{n-1}(X,A)\oplus C_{n-2}(X,A)$, we can denote it by a 3x3-matrix. My computation yielded $$ \begin{pmatrix} c_n & 0 & 0 \newline 0 & c_n & 0 \newline -id & id & c_{n-1} \end{pmatrix}, $$ where $c_n$ is the $n$th differential of $C_*(X,A)$. In the book, however, I find $$ \begin{pmatrix} c_n & 0 & (-1)^n \newline 0 & c_n & (-1)^{n-1} \newline -id & id & c_{n-1} \end{pmatrix}. $$

Question:What is meant by $(-1)^n\colon C_{n}(X,A)\to C_{n-2}(X,A)$ and how can I understand that the second matrix above is the correct expression?

**Edit:** Apparently, the correct form of the the matrix representing the $n$th differential of $C_*((X,A)\times [0,1])$ is
$$
\begin{pmatrix}
c_n & 0 & 0 \newline 0 & c_n & 0 \newline (-1)^{n+1}\cdot id & (-1)^n\cdot id & c_{n-1}
\end{pmatrix}
$$
or
$$
\begin{pmatrix}
c_n & 0 & 0 \newline 0 & c_n & 0 \newline (-1)^{n}\cdot id & (-1)^{n+1}\cdot id & c_{n-1}
\end{pmatrix},
$$
depending on the orientation of the 1-cell in $[0,1]$.