Exercise 11, § 11

Let $\Phi = 0$ be an identity between minors of an invertible square matrix $X$ of order $n$ over a commutative ring $A$, $\tilde{\Phi} = 0$ the complementary identity (Exercise 10) and $k$ an integer $> 0$. Let $Y$ be an invertible square matrix of order $n + k$ and let $\tilde{Y}_0$ be the submatrix of the adjoint matrix $\tilde{Y}$ formed by suppressing in $\tilde{Y}$ the rows of index $\leq k$ and the columns of index $\leq k$. If $\tilde{Y}_0$ is assumed to be invertible and the identity $\tilde{\Phi} = 0$ is applied to the minors of $\tilde{Y}_0$ and then each minor which appears in this identity (considered as a minor of $\tilde{Y}$) is replaced by its expression as a function of the minors of $Y$, using identity (1) of Exercise 9, an identity $\Phi_k = 0$ is obtained between minors of the matrix $Y$ (valid if $Y$ and $\tilde{Y}_0$ are invertible) which is called the extension of order $k$ of the identity $\Phi = 0$.

In particular, Let $A = (\alpha_{ij})$ be an invertible square matrix of order $n + k$, $B$ the submatrix of $A$ of order $k$ obtained by suppressing in $A$ the rows and columns of index $> k$ and $\Delta_{ij}$ the determinant of the matrix of order $k + 1$ obtained by suppressing in $A$ the rows of index $> k$ except that of index $k + i$ and the columns of index $> k$ except that of index $k + j$; if $C$ denotes the matrix $(\Delta_{ij})$ of order $n$, show the identity

valid when $A$ and $B$ are invertible (show that it is the extension of order $k$ of the total expansion of a determinant of order $n$).

State the identities obtained by extending the Laplace expansion (§ 8, no. 6) and the identity of Exercise 2 of § 8.