It's really easy question but I think I'm missing some points.
Let $F_n$ is $n$ by $n$ matrix $F_n=\det \begin{pmatrix} 1 & -1 & & & & \\ 1 & 1 & -1 & & & \\ & 1 & 1 & -1 & & \\ & & . & . & . \\ & & & 1 & 1 \end{pmatrix}$
Then $F_n=a_{11}c_{11}+a_{12}c_{12}=F_{n-1}+F_{n-2}$
where $a_{ij}$ is $(i,j)$ element of the matrix and $c_{ij}$ is cofactor.
I can't understand the second entry $F_{n-2}$. I think $a_{12}c_{12}$ is not $F_{n-2}$ since it doesn't have $1$ in $(2,1)$ entry of the matrix.