The following identity involving determinants essentially appears in E.L. Ince's book on Ordinary Differential Equations:

Let $A$ be an $n \times n$ matrix, $n \geq 3$. Denote by $A_{j_1,\ldots,j_r}^{k_1,\ldots,k_r}$ the $(n-r) \times (n-r)$ matrix obtained from $A$ by
erasing the $j_1$-th, ..., $j_r$-th row and the $k_1$-th, ..., $k_r$-th column. Then,
$$
\left|A \right| \left|A_{n-1,n}^{1,n} \right| = \left|A_{n-1}^1 \right|\left|A_n^n \right| - \left|A_{n-1}^n \right|\left|A_n^1 \right|.
$$
Any ideas of how to prove it? Is this a special case of a more general identity?