Lemma 10.15.5. Let $R$ be a ring. Let $n \geq m$. Let $A$ be an $n \times m$ matrix with coefficients in $R$. Let $J \subset R$ be the ideal generated by the $m \times m$ minors of $A$.
For any $f \in J$ there exists a $m \times n$ matrix $B$ such that $BA = f 1_{m \times m}$.
If $f \in R$ and $BA = f 1_{m \times m}$ for some $m \times n$ matrix $B$, then $f^ m \in J$.
Proof. For $I \subset \{ 1, \ldots , n\} $ with $|I| = m$, we denote by $E_ I$ the $m \times n$ matrix of the projection
and set $A_ I = E_ I A$, i.e., $A_ I$ is the $m \times m$ matrix whose rows are the rows of $A$ with indices in $I$. Let $B_ I$ be the adjugate (transpose of cofactor) matrix to $A_ I$, i.e., such that $A_ I B_ I = B_ I A_ I = \det (A_ I) 1_{m \times m}$. The $m \times m$ minors of $A$ are the determinants $\det A_ I$ for all the $I \subset \{ 1, \ldots , n\} $ with $|I| = m$. If $f \in J$ then we can write $f = \sum c_ I \det (A_ I)$ for some $c_ I \in R$. Set $B = \sum c_ I B_ I E_ I$ to see that (1) holds.
If $f 1_{m \times m} = BA$ then by the Cauchy-Binet formula (72) we have $f^ m = \sum b_ I \det (A_ I)$ where $b_ I$ is the determinant of the $m \times m$ matrix whose columns are the columns of $B$ with indices in $I$. $\square$
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Comment #3577 by Herman Rohrbach on
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