Exercise 10, § 11

From every identity $\Phi = 0$ between minors of an arbitrary invertible square matrix $X$ of order $n$ over A another identity $\tilde{\Phi} = 0$ can be derived called the complement of $\Phi = 0$, by applying the identity $\Phi = 0$ to the minors of the adjoint $\tilde{X}$ of $X$ and then replacing the minors of $\tilde{X}$ by functions of the minors of $X$ using identity (1) of Exercise 9. In this way show the following identity:

where $X^{ij, hk}$ denotes the minor of $X$ of order $n - 2$ obtained by suppressing in $X$ the rows of index $i, j$ and the columns of index $h, k$.