at.algebraic topology - Proving homotopy invariance of cellular homology by constructing a chain homotopy

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 $hcolon (X,A)times [0,1]to (Y,B)$ 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 $(-1)^ncolon C_{n}(X,A)to C_{n-2}(X,A)$ and how can I
understand that the second matrix above is the correct expression?

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