In relation to the $2\times 2$ (say) determinant $$\begin{vmatrix}a&b\\c&d\end{vmatrix},$$ consider the “bordered” determinant $$\begin{vmatrix} x&a&b\\ x&a&b\\ y&c&d \end{vmatrix}.$$ It must be zero since the first two rows are equal. Using the Laplace cofactor expansion along the first row, we obtain $$\begin{vmatrix}a&b\\c&d\end{vmatrix}x=\begin{vmatrix}x&b\\y&d\end{vmatrix}a+\begin{vmatrix}a&x\\c&y\end{vmatrix}b.\tag{1}$$ A similar argument shows $$\begin{vmatrix}a&b\\c&d\end{vmatrix}y=\begin{vmatrix}x&b\\y&d\end{vmatrix}c+\begin{vmatrix}a&x\\c&y\end{vmatrix}d.\tag{2}$$ Combining (1) and (2) in vector form, we have $$\begin{vmatrix}a&b\\c&d\end{vmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{vmatrix}x&b\\y&d\end{vmatrix}\begin{bmatrix}a\\c\end{bmatrix}+\begin{vmatrix}a&x\\c&y\end{vmatrix}\begin{bmatrix}b\\d\end{bmatrix}.\tag{3}$$ Notice also that if we take $x=1$ and $y=0$ in (1), we obtain the cofactor expansion of $\begin{vmatrix}a&b\\c&d\end{vmatrix}$ along the first row, and if we take $x=0$ and $y=1$ in (2), we obtain the cofactor expansion along the second row.
These ideas generalize to $n\times n$ matrices. In fact if $E$ is an $n$-dimensional vector space and $\Delta$ is a determinant function in $E$, then using an $(n+1)$-dimensional cofactor expansion we can prove $$\Delta(x_1,\ldots,x_n)x=\sum_{j=1}^n\Delta(x_1,\ldots,x,\ldots,x_n)x_j,\tag{4}$$ where $x$ is in the $j$-th position of $\Delta$ on the right. Applying projection operators on both sides of (4) we can recover the $n$-dimensional cofactor expansions.
Does anyone know of a classical source or name for this result? I’ve looked in a number of places including thumbing through Muir and Metzler but haven’t found it. Given the close connection to Laplace, I’ve been calling it a “vector-valued” Laplace expansion, but I’m not sure if it has another name.