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• Let $\mathbb{K}$ be a commutative ring. All matrices that appear in the following are matrices over $\mathbb{K}$.

• Let $\mathbb{N}=\left\{ 0,1,2,\ldots\right\} $.

• For every $n\in\mathbb{N}$, we let $\left[ n\right] $ denote the set $\left\{ 1,2,\ldots,n\right\} $.

• Fix $n\in\mathbb{N}$.

• Let $S_{n}$ denote the $n$-th symmetric group (i.e., the group of permutations of $\left[ n\right] $).

• If $A\in\mathbb{K}^{n\times n}$ is an $n\times n$-matrix, and if $I$ and $J$ are two subsets of $\left[ n\right] $, then $A_{J}^{I}$ is the $\left\vert I\right\vert \times\left\vert J\right\vert $-matrix defined as follows: Write $A$ in the form $A=\left( a_{i,j}\right) _{1\leq i\leq n,\ 1\leq j\leq m}$; write the set $I$ in the form $I=\left\{ i_{1}<i_{2}<\cdots<i_{u}\right\} $; write the set $J$ in the form $J=\left\{ j_{1}<j_{2}<\cdots<j_{v}\right\} $. Then, set $A_{J}^{I}=\left( a_{i_{x},j_{y}}\right) _{1\leq x\leq u,\ 1\leq y\leq v}$. (Thus, roughly speaking, $A_{J}^{I}$ is the $\left\vert I\right\vert \times\left\vert J\right\vert $-matrix obtained from $A$ by removing all rows whose indices do not belong to $I$, and removing all columns whose indices do not belong to $J$.)

Theorem 1 (Jacobi's complementary minor formula). Let $A\in\mathbb{K} ^{n\times n}$ be an invertible $n\times n$-matrix. Let $I$ and $J$ be two subsets of $\left[ n\right] $ such that $\left\vert I\right\vert =\left\vert J\right\vert $. Then,

$\det\left( A_{J}^{I}\right) =\left( -1\right) ^{\sum I+\sum J}\det A\cdot\det\left( \left( A^{-1}\right) _{\widetilde{I}}^{\widetilde{J} }\right) $.

• Theorem 1 is Lemma A.1 (e) in Sergio Caracciolo, Alan D. Sokal, Andrea Sportiello, Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians, arXiv:1105.6270v2 (published in: Advances in Applied Mathematics 50, 474--594 (2013)). In the paragraph following Theorem A.16, a proof is given using what the authors call "Grassmann-Berezin integration" (despite its name, a purely algebraic mock-calculus on the exterior algebra of a vector space).

• Theorem 1 is (1) in Pierre Lalonde, A non-commutative version of Jacobi's equality on the cofactors of a matrix, Discrete Mathematics 158 (1996), pp. 161--172. The goal of the paper is to generalize it to a (mildly) noncommutative setting.

• Theorem 1 is Additional exercise 19 in my Notes on the combinatorial fundamentals of algebra, version of 12 December 2016. The first proof I give is fairly classical and similar to the ones given by Federico Poloni and Denis Serre, but requires no WLOG assumptions (instead of using the Schur complement, I use a cheap generalization of it which is Exercise 50 in the notes). The formal bookkeeping that leads to the sign $\left( -1\right) ^{\sum I+\sum J}$ takes a huge lot of space, although it is so easy to convince yourself of it with some handwaving that you might not notice that it requires any proof at all. The second proof is an expansion of an argument briefly outlined in D. Laksov, A. Lascoux, P. Pragacz, and A. Thorup, The LLPT Notes (Chapter SCHUR, proof of (1.9)).

Note that every source uses different notations. What I call $A_{J}^{I}$ above is called $A_{IJ}$ in the paper by Caracciolo, Sokal and Sportiello, is called $A\left[ I,J\right] $ in Lalonde's paper, and is called $\operatorname*{sub}\nolimits_{w\left( I\right) }^{w\left( J\right) }A$ in my notes. Also, the $I$ and $J$ in the paper by Caracciolo, Sokal and Sportiello correspond to the $\widetilde{I}$ and $\widetilde{J}$ in Theorem 1 above.

This proof would become a lot shorter if I didn't care for the signs and would only prove the weaker claim that $\det\left( A_{J}^{I}\right) = \pm \det A\cdot\det\left( \left( A^{-1}\right) _{\widetilde{I}}^{\widetilde{J} }\right) $ for some value of $\pm$. But this weaker claim is not as useful as Theorem 1 in its full version (in particular, it would not suffice to fill the gap in Macdonald's book that has motivated this question).

If $K$ is a subset of $\left[ n\right] $, and if $k = \left|K\right|$, then we let $w\left( K\right) $ be the (unique) permutation $\sigma\in S_{n}$ whose first $k$ values $\sigma\left( 1\right) ,\sigma\left( 2\right) ,\ldots,\sigma\left( k\right) $ are the elements of $K$ in increasing order, and whose next $n-k$ values $\sigma\left( k+1\right) ,\sigma\left( k+2\right) ,\ldots ,\sigma\left( n\right) $ are the elements of $\widetilde{K}$ in increasing order.

Lemma 2. Let $K$ be a subset of $\left[ n\right] $. Then, $\left( -1\right) ^{w\left( K\right) }=\left( -1\right) ^{\sum K-\left( 1+2+\cdots+\left\vert K\right\vert \right) }$.

Proof of Lemma 2. Let $k=\left\vert K\right\vert $. Let $a_{1},a_{2} ,\ldots,a_{k}$ be the $k$ elements of $K$ in increasing order (with no repetitions). Let $b_{1},b_{2},\ldots,b_{n-k}$ be the $n-k$ elements of $\widetilde{K}$ in increasing order (with no repetitions). Let $\gamma =w\left( K\right) $. Then, the definition of $w\left( K\right) $ shows that the first $k$ values $\gamma\left( 1\right) ,\gamma\left( 2\right) ,\ldots,\gamma\left( k\right) $ of $\gamma$ are the elements of $K$ in increasing order (that is, $a_{1},a_{2},\ldots,a_{k}$), and the next $n-k$ values $\gamma\left( k+1\right) ,\gamma\left( k+2\right) ,\ldots ,\gamma\left( n\right) $ of $\gamma$ are the elements of $\widetilde{K}$ in increasing order (that is, $b_{1},b_{2},\ldots,b_{n-k}$). In other words,

$\left( \gamma\left( 1\right) ,\gamma\left( 2\right) ,\ldots ,\gamma\left( n\right) \right) =\left( a_{1},a_{2},\ldots,a_{k} ,b_{1},b_{2},\ldots,b_{n-k}\right) $.

At the end, the first $k$ positions of the list are filled with $a_{1} ,a_{2},\ldots,a_{k}$ (in this order), whereas the remaining $n-k$ positions are filled with the remaining entries $b_{1},b_{2},\ldots,b_{n-k}$ (again, in this order, because the switches have never disrupted their strictly-increasing relative order). Thus, at the end, your list is precisely $\left( a_{1},a_{2},\ldots,a_{k},b_{1},b_{2},\ldots,b_{n-k}\right) =\left( \gamma\left( 1\right) ,\gamma\left( 2\right) ,\ldots,\gamma\left( n\right) \right) $. You have used a total of

$\left( a_{1}-1\right) +\left( a_{2}-2\right) +\cdots+\left( a_{k}-k\right) $

$=\underbrace{\left( a_{1}+a_{2}+\cdots+a_{k}\right) }_{\substack{=\sum K\\\text{(by the definition of }a_{1},a_{2},\ldots,a_{k}\text{)} }}-\underbrace{\left( 1+2+\cdots+k\right) }_{\substack{=1+2+\cdots +\left\vert K\right\vert \\\text{(since }k=\left\vert K\right\vert \text{)}}}$

$=\sum K-\left( 1+2+\cdots+\left\vert K\right\vert \right) $

switches. Thus, you have obtained the list $\left( \gamma\left( 1\right) ,\gamma\left( 2\right) ,\ldots,\gamma\left( n\right) \right) $ from the list $\left( 1,2,\ldots,n\right) $ by $\sum K-\left( 1+2+\cdots+\left\vert K\right\vert \right) $ switches of adjacent entries. In other words, the permutation $\gamma$ is a composition of $\sum K-\left( 1+2+\cdots+\left\vert K\right\vert \right) $ simple transpositions (where a "simple transposition" means a transposition switching $u$ with $u+1$ for some $u$). Hence, it has sign $\left( -1\right) ^{\sum K-\left( 1+2+\cdots+\left\vert K\right\vert \right) }$. This proves Lemma 2.

If $k\in\mathbb{N}$ and if $V$ is a $\mathbb{K}$-module, then $\wedge^{k}V$ shall mean the $k$-th exterior power of $V$. If $k\in\mathbb{N}$, if $V$ and $W$ are two $\mathbb{K}$-modules, and if $f:V\rightarrow W$ is a $\mathbb{K} $-linear map, then the $\mathbb{K}$-linear map $\wedge^{k}V\rightarrow \wedge^{k}W$ canonically induced by $f$ will be denoted by $\wedge^{k}f$. It is well-known that if $V$ and $W$ are two $\mathbb{K}$-modules, if $f:V\rightarrow W$ is a $\mathbb{K}$-linear map, then

(1) $\left( \wedge^{k}f\right) \left( a\right) \cdot\left( \wedge^{\ell}f\right) \left( b\right) =\left( \wedge^{k+\ell}f\right) \left( ab\right) $

for any $k\in\mathbb{N}$, $\ell\in\mathbb{N}$, $a\in\wedge^{k}V$ and $b\in\wedge^{\ell}V$.

If $V$ is a $\mathbb{K}$-module, then

(2) $uv=\left( -1\right) ^{k\ell}vu$

for any $k\in\mathbb{N}$, $\ell\in\mathbb{N}$, $u\in\wedge^{k}V$ and $v\in\wedge^{\ell}V$.

Proposition 2a. Let $f:\mathbb{K}^{n}\rightarrow\mathbb{K}^{n}$ be a $\mathbb{K}$-linear map. The map $\wedge^{n}f:\wedge^{n}\left( \mathbb{K} ^{n}\right) \rightarrow\wedge^{n}\left( \mathbb{K}^{n}\right) $ is multiplication by $\det f$. In other words, every $z\in\wedge^{n}\left( \mathbb{K}^{n}\right) $ satisfies

(3) $\left( \wedge^{n}f\right) \left( z\right) =\left( \det f\right) z$.

Let $\left( e_{1},e_{2},\ldots,e_{n}\right) $ be the standard basis of the $\mathbb{K}$-module $\mathbb{K}^{n}$. (Thus, $e_{i}$ is the column vector whose $i$-th entry is $1$ and whose all other entries are $0$.)

For every subset $K$ of $\left[ n\right] $, we define $e_{K}\in \wedge^{\left\vert K\right\vert }\left( \mathbb{K}^{n}\right) $ to be the element $e_{k_{1}}\wedge e_{k_{2}}\wedge\cdots\wedge e_{k_{\left\vert K\right\vert }}$, where $K$ is written in the form $K=\left\{ k_{1} <k_{2}<\cdots<k_{\left\vert K\right\vert }\right\} $.

It is well-known that, for every $k\in\mathbb{N}$, the family $\left( e_{K}\right) _{K\in\mathcal{P}_{k}\left( \left[ n\right] \right) }$ is a basis of the $\mathbb{K}$-module $\wedge^{k}\left( \mathbb{K}^{n}\right) $. Applying this to $k=n$, we conclude that the family $\left( e_{K}\right) _{K\in\mathcal{P}_{n}\left( \left[ n\right] \right) }$ is a basis of the $\mathbb{K}$-module $\wedge^{n}\left( \mathbb{K}^{n}\right) $. Since this family $\left( e_{K}\right) _{K\in\mathcal{P}_{n}\left( \left[ n\right] \right) }$ is the one-element family $\left( e_{\left[ n\right] }\right) $ (because the only $K\in\mathcal{P}_{n}\left( \left[ n\right] \right) $ is the set $\left[ n\right] $), this rewrites as follows: The one-element family $\left( e_{\left[ n\right] }\right) $ is a basis of the $\mathbb{K}$-module $\wedge^{n}\left( \mathbb{K}^{n}\right) $.

If $B$ is an $n\times n$-matrix and $k\in\mathbb{N}$, then evaluating the map $\wedge^{k}f_{B}$ on the elements of the basis $\left( e_{K}\right) _{K\in\mathcal{P}_{k}\left( \left[ n\right] \right) }$ of $\wedge ^{k}\left( \mathbb{K}^{n}\right) $, and expanding the results again in this basis gives rise to coefficients which are the $k\times k$-minors of $B$. More precisely:

Proposition 3. Let $B\in\mathbb{K}^{n\times n}$, $k\in\mathbb{N}$ and $J\in\mathcal{P}_{k}\left( \left[ n\right] \right) $. Then,

$\left( \wedge^{k}f_{B}\right) \left( e_{J}\right) = \sum\limits_{I\in\mathcal{P}_{k}\left( \left[ n\right] \right) }\det\left( B_{J} ^{I}\right) e_{I}$.

(This generalizes: If $u\in\mathbb{N}$, $v \in \mathbb{N}$, $B\in\mathbb{K}^{u\times v}$, $k\in\mathbb{N}$ and $J\in\mathcal{P}_{k}\left( \left[ v\right] \right) $, then $\left( \wedge^{k}f_{B}\right) \left( e_{J}\right) = \sum\limits_{I\in\mathcal{P}_{k}\left( \left[ u\right] \right) }\det\left( B_{J} ^{I}\right) e_{I}$, where the elements $e_{J}\in\wedge^{k}\left( \mathbb{K}^{v}\right) $ and $e_{I}\in\wedge^{k}\left( \mathbb{K}^{u}\right) $ are defined as before but with $v$ and $u$ instead of $n$.)

Lemma 4. Let $K$ be a subset of $\left[ n\right] $. Then,

$e_{K}e_{\widetilde{K}}=\left( -1\right) ^{\sum K-\left( 1+2+\cdots+\left\vert K\right\vert \right) }e_{\left[ n\right] }$.

Proof of Lemma 4. Let $k = \left|K\right|$. Let $\sigma$ be the permutation $w\left( K\right) \in S_{n}$ defined above. Its first $k$ values $\sigma\left( 1\right) ,\sigma\left( 2\right) ,\ldots,\sigma\left( k\right) $ are the elements of $K$ in increasing order; thus, $e_{\sigma\left( 1\right) }\wedge e_{\sigma\left( 2\right) }\wedge\cdots\wedge e_{\sigma\left( k\right) }=e_{K}$. Its next $n-k$ values $\sigma\left( k+1\right) ,\sigma\left( k+2\right) ,\ldots,\sigma\left( n\right) $ are the elements of $\widetilde{K}$ in increasing order; thus, $e_{\sigma\left( k+1\right) }\wedge e_{\sigma\left( k+2\right) }\wedge\cdots\wedge e_{\sigma\left( n\right) }=e_{\widetilde{K}}$.

From $\sigma=w\left( K\right) $, we obtain $\left( -1\right) ^{\sigma }=\left( -1\right) ^{w\left( K\right) }=\left( -1\right) ^{\sum K-\left( 1+2+\cdots+\left\vert K\right\vert \right) }$ (by Lemma 2).

Now, it is well-known that

$e_{\sigma\left( 1\right) }\wedge e_{\sigma\left( 2\right) }\wedge \cdots\wedge e_{\sigma\left( n\right) }=\left( -1\right) ^{\sigma }\underbrace{e_{1}\wedge e_{2}\wedge\cdots\wedge e_{n}}_{=e_{\left[ n\right] }}=\left( -1\right) ^{\sigma}e_{\left[ n\right] }$.

Hence,

$\left( -1\right) ^{\sigma}e_{\left[ n\right] }=e_{\sigma\left( 1\right) }\wedge e_{\sigma\left( 2\right) }\wedge\cdots\wedge e_{\sigma\left( n\right) }$

$=\underbrace{\left( e_{\sigma\left( 1\right) }\wedge e_{\sigma\left( 2\right) }\wedge\cdots\wedge e_{\sigma\left( k\right) }\right) }_{=e_{K} }\underbrace{\left( e_{\sigma\left( k+1\right) }\wedge e_{\sigma\left( k+2\right) }\wedge\cdots\wedge e_{\sigma\left( n\right) }\right) }_{=e_{\widetilde{K}}}=e_{K}e_{\widetilde{K}}$.

Since $\left( -1\right) ^{\sigma}=\left( -1\right) ^{\sum K-\left( 1+2+\cdots+\left\vert K\right\vert \right) }$, this rewrites as $\left( -1\right) ^{\sum K-\left( 1+2+\cdots+\left\vert K\right\vert \right) }e_{\left[ n\right] }= e_K e_{\widetilde{K}}$. This proves Lemma 4.

Corollary 5. Let $B\in\mathbb{K}^{n\times n}$, $k\in\mathbb{N}$ and $J\in\mathcal{P}_{k}\left( \left[ n\right] \right) $. Then, every $K\in\mathcal{P}_{k}\left( \left[ n\right] \right) $ satisfies

$\left( \wedge^{k}f_{B}\right) \left( e_{J}\right) e_{\widetilde{K} }=\left( -1\right) ^{\sum K-\left( 1+2+\cdots+k\right) }\det\left( B_{J}^{K}\right) e_{\left[ n\right] }$.

Now, forget that we fixed $I$. We thus have proven that

$e_{I}e_{\widetilde{K}}=0$ for every $I\in\mathcal{P}_{k}\left( \left[ n\right] \right) $ satisfying $I\neq K$.

Hence,

(4) $\sum\limits_{\substack{I\in\mathcal{P}_{k}\left( \left[ n\right] \right) ;\\I\neq K}}\det\left( B_{J}^{I}\right) \underbrace{e_{I}e_{\widetilde{K}} }_{=0}=0$.

Proposition 3 yields

$\left( \wedge^{k}f_{B}\right) \left( e_{J}\right) =\sum\limits_{I\in \mathcal{P}_{k}\left( \left[ n\right] \right) }\det\left( B_{J} ^{I}\right) e_{I}$.

Multiplying both sides of this equality by $e_{\widetilde{K}}$ from the right, we find

$\left( \wedge^{k}f_{B}\right) \left( e_{J}\right) e_{\widetilde{K}} =\sum\limits_{I\in\mathcal{P}_{k}\left( \left[ n\right] \right) }\det\left( B_{J}^{I}\right) e_{I}e_{\widetilde{K}}$

$=\det\left( B_{J}^{K}\right) e_{K}e_{\widetilde{K}}+\sum\limits_{\substack{I\in \mathcal{P}_{k}\left( \left[ n\right] \right) ;\\I\neq K}}\det\left( B_{J}^{I}\right) e_{I}e_{\widetilde{K}}$

$=\det\left( B_{J}^{K}\right) \underbrace{e_{K}e_{\widetilde{K}} }_{\substack{=\left( -1\right) ^{\sum K-\left( 1+2+\cdots+\left\vert K\right\vert \right) }e_{\left[ n\right] }\\\text{(by Lemma 4)}}}$ (by (4))

$=\left( -1\right) ^{\sum K-\left( 1+2+\cdots+\left\vert K\right\vert \right) }\det\left( B_{J}^{K}\right) e_{\left[ n\right] }$

$=\left( -1\right) ^{\sum K-\left( 1+2+\cdots+k\right) }\det\left( B_{J}^{K}\right) e_{\left[ n\right] }$

(since $\left\vert K\right\vert =k$). This proves Corollary 5.

Proof of Theorem 1. Set $k=\left\vert I\right\vert =\left\vert J\right\vert $. Notice that $\left\vert \widetilde{I}\right\vert =n-k$ (since $\left\vert I\right\vert =k$) and $\left\vert \widetilde{J}\right\vert =n-k$ (similarly).

The maps $f_{A}$ and $f_{A^{-1}}$ are mutually inverse (since the map $\mathbb{K}^{n\times n}\rightarrow\operatorname*{End}\left( \mathbb{K} ^{n}\right) ,\ B\mapsto f_{B}$ is a ring homomorphism). Hence, the maps $\wedge^{n-k}f_{A}$ and $\wedge^{n-k}f_{A^{-1}}$ are mutually inverse (since $\wedge^{n-k}$ is a functor). Thus, $\left( \wedge^{n-k}f_{A}\right) \circ\left( \wedge^{n-k}f_{A^{-1}}\right) =\operatorname*{id}$. Now, $y=\left( \wedge^{n-k}f_{A^{-1}}\right) \left( e_{\widetilde{I}}\right) $, so that

$\left( \wedge^{n-k}f_{A}\right) \left( y\right) =\left( \wedge ^{n-k}f_{A}\right) \left( \left( \wedge^{n-k}f_{A^{-1}}\right) \left( e_{\widetilde{I}}\right) \right) =\underbrace{\left( \left( \wedge ^{n-k}f_{A}\right) \circ\left( \wedge^{n-k}f_{A^{-1}}\right) \right) }_{=\operatorname*{id}}\left( e_{\widetilde{I}}\right) =e_{\widetilde{I}}$.

But (1) (applied to $V=\mathbb{K}^{n}$, $W=\mathbb{K}^{n}$, $f=f_{A}$, $\ell=n-k$, $a=e_{J}$ and $b=y$) yields

$\left( \wedge^{k}f_{A}\right) \left( e_{J}\right) \cdot\left( \wedge^{n-k}f_{A}\right) \left( y\right) =\left( \wedge^{n}f_{A}\right) \left( e_{J}y\right) $.

Thus,

$\left( \wedge^{n}f_{A}\right) \left( e_{J}y\right) =\left( \wedge ^{k}f_{A}\right) \left( e_{J}\right) \cdot\underbrace{\left( \wedge ^{n-k}f_{A}\right) \left( y\right) }_{=e_{\widetilde{I}}}$

$=\left( \wedge^{k}f_{A}\right) \left( e_{J}\right) e_{\widetilde{I}} = \left( -1\right) ^{\sum I-\left( 1+2+\cdots+k\right) }\det\left( A_{J}^{I}\right) e_{\left[ n\right] }$

(by Corollary 5, applied to $B=A$ and $K=I$).

Hence,

$\left( -1\right) ^{\sum I-\left( 1+2+\cdots+k\right) }\det\left( A_{J}^{I}\right) e_{\left[ n\right] }$

$=\left( \wedge^{n}f_{A}\right) \left( e_{J}y\right) =\underbrace{\left( \det f_{A}\right) }_{=\det A}e_{J}y$

(by (3), applied to $f=f_{A}$ and $z=e_{J}y$)

(5) $=\left( \det A\right) e_{J}y$.

But (2) (applied to $\ell=n-k$, $u=e_{J}$ and $v=y$) yields

$e_{J} y = \left(-1\right)^{k \left(n-k\right)} \underbrace{y}_{=\left( \wedge^{n-k}f_{A^{-1}}\right) \left( e_{\widetilde{I}}\right) } \underbrace{e_{J}} _{=e_{\widetilde{\widetilde{J}}}}$

$=\left( -1\right) ^{k\left( n-k\right) }\underbrace{\left( \wedge ^{n-k}f_{A^{-1}}\right) \left( e_{\widetilde{I}}\right) e_{\widetilde{\widetilde{J}}}}_{\substack{=\left( -1\right) ^{\sum \widetilde{J}-\left( 1+2+\cdots+\left( n-k\right) \right) }\det\left( \left( A^{-1}\right) _{\widetilde{I}}^{\widetilde{J}}\right) e_{\left[ n\right] }\\\text{(by Corollary 5, applied to }A^{-1}\text{, }n-k\text{, }\widetilde{I}\text{ and }\widetilde{J}\\\text{instead of }B\text{, }k\text{, }J\text{ and }K\text{)}}}$

$=\left( -1\right) ^{k\left( n-k\right) }\left( -1\right) ^{\sum \widetilde{J}-\left( 1+2+\cdots+\left( n-k\right) \right) }\det\left( \left( A^{-1}\right) _{\widetilde{I}}^{\widetilde{J}}\right) e_{\left[ n\right] }$

(6) $=\left( -1\right) ^{k\left( n-k\right) +\sum\widetilde{J}-\left( 1+2+\cdots+\left( n-k\right) \right) }\det\left( \left( A^{-1}\right) _{\widetilde{I}}^{\widetilde{J}}\right) e_{\left[ n\right] }$.

But

$k\left( n-k\right) +\underbrace{\sum\widetilde{J}}_{=\sum\left\{ 1,2,\ldots,n\right\} -\sum J}-\underbrace{\left( 1+2+\cdots+\left( n-k\right) \right) }_{=\sum\left\{ 1,2,\ldots,n-k\right\} }$

$=k\left( n-k\right) +\sum\left\{ 1,2,\ldots,n\right\} -\sum J-\sum\left\{ 1,2,\ldots,n-k\right\} $

$=\underbrace{\sum\left\{ 1,2,\ldots,n\right\} -\sum\left\{ 1,2,\ldots ,n-k\right\} }_{\substack{=\sum\left\{ n-k+1,n-k+2,\ldots,n\right\} \\=k\left( n-k\right) +\sum\left\{ 1,2,\ldots,k\right\} \\=k\left( n-k\right) +\left( 1+2+\cdots+k\right) }}+k\left( n-k\right) -\sum J$

$=2k\left( n-k\right) +\left( 1+2+\cdots+k\right) -\sum J$

$\equiv-\left( 1+2+\cdots+k\right) -\sum J\operatorname{mod}2$.

Hence,

$\left( -1\right) ^{k\left( n-k\right) +\sum\widetilde{J}-\left( 1+2+\cdots+\left( n-k\right) \right) }=\left( -1\right) ^{-\left( 1+2+\cdots+k\right) -\sum J}$.

Thus, (6) rewrites as

$e_{J}y=\left( -1\right) ^{-\left( 1+2+\cdots+k\right) -\sum J}\det\left( \left( A^{-1}\right) _{\widetilde{I}}^{\widetilde{J}}\right) e_{\left[ n\right] }$.

Hence, (5) rewrites as

$\left( -1\right) ^{\sum I-\left( 1+2+\cdots+k\right) }\det\left( A_{J}^{I}\right) e_{\left[ n\right] }$

$=\left( \det A\right) \left( -1\right) ^{-\left( 1+2+\cdots+k\right) -\sum J}\det\left( \left( A^{-1}\right) _{\widetilde{I}}^{\widetilde{J} }\right) e_{\left[ n\right] }$.

We can "cancel" $e_{\left[ n\right] }$ from this equality (because if $\lambda$ and $\mu$ are two elements of $\mathbb{K}$ satisfying $\lambda e_{\left[ n\right] }=\mu e_{\left[ n\right] }$, then $\lambda=\mu$), and thus obtain

$\left( -1\right) ^{\sum I-\left( 1+2+\cdots+k\right) }\det\left( A_{J}^{I}\right) $

$=\left( \det A\right) \left( -1\right) ^{-\left( 1+2+\cdots+k\right) -\sum J}\det\left( \left( A^{-1}\right) _{\widetilde{I}}^{\widetilde{J} }\right) $.

Dividing this equality by $\left( -1\right) ^{\sum I-\left( 1+2+\cdots +k\right) }$, we obtain

$\det\left( A_{J}^{I}\right) $

$=\left(\det A\right) \dfrac{\left( -1\right) ^{-\left( 1+2+\cdots+k\right) -\sum J}}{\left( -1\right) ^{\sum I-\left( 1+2+\cdots+k\right) }}\det\left( \left( A^{-1}\right) _{\widetilde{I}}^{\widetilde{J}}\right) $

$=\left( -1\right) ^{\sum I+\sum J}\det A\cdot\det\left( \left( A^{-1}\right) _{\widetilde{I}}^{\widetilde{J}}\right) $.

This proves Theorem 1.

• Let $\mathbb{K}$ be a commutative ring. All matrices that appear in the following are matrices over $\mathbb{K}$.

• Let $\mathbb{N}=\left\{ 0,1,2,\ldots\right\} $.

• For every $n\in\mathbb{N}$, we let $\left[ n\right] $ denote the set $\left\{ 1,2,\ldots,n\right\} $.

• Fix $n\in\mathbb{N}$.

• Let $S_n$ denote the $n$-th symmetric group (i.e., the group of permutations of $\left[ n\right] $).

• If $A\in\mathbb{K}^{n\times m}$ is an $n\times m$-matrix, if $I$ is a subset of $\left[ n \right]$, and if $J$ is a subset of $\left[ m \right]$, then $A_J^I$ is the $\left| I\right| \times\left| J\right| $-matrix defined as follows: Write $A$ in the form $A=\left( a_{i,j}\right) _{1\leq i\leq n,\ 1\leq j\leq m}$; write the set $I$ in the form $I = \left\{ i_1 < i_2 < \cdots < i_u \right\}$; write the set $J$ in the form $J = \left\{ j_1 < j_2 < \cdots < j_v \right\}$. Then, set $A_J^I = \left( a_{i_x, j_y} \right) _{1\leq x\leq u,\ 1\leq y\leq v}$. (Thus, roughly speaking, $A_J^I$ is the $\left| I\right| \times\left| J\right| $-matrix obtained from $A$ by removing all rows whose indices do not belong to $I$, and removing all columns whose indices do not belong to $J$.)

Theorem 1 (Jacobi's complementary minor formula). Let $A\in\mathbb{K} ^{n\times n}$ be an invertible $n\times n$-matrix. Let $I$ and $J$ be two subsets of $\left[ n\right] $ such that $\left| I\right| =\left| J\right| $. Then, \begin{align} \det\left( A_{J}^{I}\right) =\left( -1\right) ^{\sum I+\sum J}\det A\cdot\det\left( \left( A^{-1}\right) _{\widetilde{I}}^{\widetilde{J} }\right) . \end{align}

• Theorem 1 is Lemma A.1 (e) in Sergio Caracciolo, Alan D. Sokal, Andrea Sportiello, Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians, arXiv:1105.6270v2 (published in: Advances in Applied Mathematics 50, 474--594 (2013)). In the paragraph following Theorem A.16, a proof is given using what the authors call "Grassmann-Berezin integration" (despite its name, a purely algebraic mock-calculus on the exterior algebra of a vector space).

• Theorem 1 is (1) in Pierre Lalonde, A non-commutative version of Jacobi's equality on the cofactors of a matrix, Discrete Mathematics 158 (1996), pp. 161--172. The goal of the paper is to generalize it to a (mildly) noncommutative setting.

• Theorem 1 is Exercise 6.56 in my Notes on the combinatorial fundamentals of algebra, version of 10 January 2019. The first proof I give is fairly classical and similar to the ones given by Federico Poloni and Denis Serre, but requires no WLOG assumptions (instead of using the Schur complement, I use a cheap generalization of it which is Exercise 6.38 in the notes). The formal bookkeeping that leads to the sign $\left( -1\right) ^{\sum I+\sum J}$ takes a huge lot of space, although it is so easy to convince yourself of it with some handwaving that you might not notice that it requires any proof at all. The second proof is an expansion of an argument briefly outlined in D. Laksov, A. Lascoux, P. Pragacz, and A. Thorup, The LLPT Notes, old (2001) version (Chapter SCHUR, proof of (1.9)).

Note that every source uses different notations. What I call $A_J^I$ above is called $A_{IJ}$ in the paper by Caracciolo, Sokal and Sportiello, is called $A\left[ I,J\right] $ in Lalonde's paper, and is called $\operatorname*{sub}\nolimits_{w\left( I\right) }^{w\left( J\right) }A$ in my notes. Also, the $I$ and $J$ in the paper by Caracciolo, Sokal and Sportiello correspond to the $\widetilde{I}$ and $\widetilde{J}$ in Theorem 1 above.

This proof would become a lot shorter if I didn't care for the signs and would only prove the weaker claim that $\det\left( A_J^I \right) = \pm \det A\cdot \det\left( \left( A^{-1}\right) _{\widetilde{I}}^{\widetilde{J} }\right) $ for some value of $\pm$. But this weaker claim is not as useful as Theorem 1 in its full version (in particular, it would not suffice to fill the gap in Macdonald's book that has motivated this question).

If $K$ is a subset of $\left[ n\right] $, and if $k = \left|K\right|$, then we let $w\left( K\right) $ be the (unique) permutation $\sigma\in S_n$ whose first $k$ values $\sigma\left( 1\right) ,\sigma\left( 2\right) ,\ldots,\sigma\left( k\right) $ are the elements of $K$ in increasing order, and whose next $n-k$ values $\sigma\left( k+1\right) ,\sigma\left( k+2\right) ,\ldots ,\sigma\left( n\right) $ are the elements of $\widetilde{K}$ in increasing order.

Lemma 2. Let $K$ be a subset of $\left[ n\right] $. Then, $\left( -1\right) ^{w\left( K\right) }=\left( -1\right) ^{\sum K-\left( 1+2+\cdots+\left| K\right| \right) }$.

Proof of Lemma 2. Let $k=\left| K\right| $. Let $a_{1},a_{2} ,\ldots,a_{k}$ be the $k$ elements of $K$ in increasing order (with no repetitions). Let $b_{1},b_{2},\ldots,b_{n-k}$ be the $n-k$ elements of $\widetilde{K}$ in increasing order (with no repetitions). Let $\gamma =w\left( K\right) $. Then, the definition of $w\left( K\right) $ shows that the first $k$ values $\gamma\left( 1\right) ,\gamma\left( 2\right) ,\ldots,\gamma\left( k\right) $ of $\gamma$ are the elements of $K$ in increasing order (that is, $a_{1},a_{2},\ldots,a_{k}$), and the next $n-k$ values $\gamma\left( k+1\right) ,\gamma\left( k+2\right) ,\ldots ,\gamma\left( n\right) $ of $\gamma$ are the elements of $\widetilde{K}$ in increasing order (that is, $b_{1},b_{2},\ldots,b_{n-k}$). In other words, \begin{align} \left( \gamma\left( 1\right) ,\gamma\left( 2\right) ,\ldots ,\gamma\left( n\right) \right) =\left( a_{1},a_{2},\ldots,a_{k} ,b_{1},b_{2},\ldots,b_{n-k}\right) . \end{align}

At the end, the first $k$ positions of the list are filled with $a_{1} ,a_{2},\ldots,a_{k}$ (in this order), whereas the remaining $n-k$ positions are filled with the remaining entries $b_{1},b_{2},\ldots,b_{n-k}$ (again, in this order, because the switches have never disrupted their strictly-increasing relative order). Thus, at the end, your list is precisely $\left( a_{1},a_{2},\ldots,a_{k},b_{1},b_{2},\ldots,b_{n-k}\right) =\left( \gamma\left( 1\right) ,\gamma\left( 2\right) ,\ldots,\gamma\left( n\right) \right) $. You have used a total of \begin{align} & \left( a_{1}-1\right) +\left( a_{2}-2\right) +\cdots+\left( a_{k}-k\right) \\ & = \underbrace{\left( a_{1}+a_{2}+\cdots+a_{k}\right) }_{\substack{=\sum K\\\text{(by the definition of }a_{1},a_{2},\ldots,a_{k}\text{)} }}-\underbrace{\left( 1+2+\cdots+k\right) }_{\substack{=1+2+\cdots +\left| K\right| \\\text{(since }k=\left| K\right| \text{)}}} \\ & =\sum K-\left( 1+2+\cdots+\left| K\right| \right) \end{align} switches. Thus, you have obtained the list $\left( \gamma\left( 1\right) ,\gamma\left( 2\right) ,\ldots,\gamma\left( n\right) \right) $ from the list $\left( 1,2,\ldots,n\right) $ by $\sum K-\left( 1+2+\cdots+\left| K\right| \right) $ switches of adjacent entries. In other words, the permutation $\gamma$ is a composition of $\sum K-\left( 1+2+\cdots+\left| K\right| \right) $ simple transpositions (where a "simple transposition" means a transposition switching $u$ with $u+1$ for some $u$). Hence, it has sign $\left( -1\right) ^{\sum K-\left( 1+2+\cdots+\left| K\right| \right) }$. This proves Lemma 2. $\blacksquare$

If $k\in\mathbb{N}$ and if $V$ is a $\mathbb{K}$-module, then $\wedge^{k}V$ shall mean the $k$-th exterior power of $V$. If $k\in\mathbb{N}$, if $V$ and $W$ are two $\mathbb{K}$-modules, and if $f:V\rightarrow W$ is a $\mathbb{K} $-linear map, then the $\mathbb{K}$-linear map $\wedge^{k}V\rightarrow \wedge^{k}W$ canonically induced by $f$ will be denoted by $\wedge^{k}f$. It is well-known that if $V$ and $W$ are two $\mathbb{K}$-modules, if $f:V\rightarrow W$ is a $\mathbb{K}$-linear map, then \begin{align} \left( \wedge^{k}f\right) \left( a\right) \cdot\left( \wedge^{\ell}f\right) \left( b\right) =\left( \wedge^{k+\ell}f\right) \left( ab\right) \label{darij1.eq1} \tag{1} \end{align} for any $k\in\mathbb{N}$, $\ell\in\mathbb{N}$, $a\in\wedge^{k}V$ and $b\in\wedge^{\ell}V$.

If $V$ is a $\mathbb{K}$-module, then \begin{align} uv=\left( -1\right) ^{k\ell}vu \label{darij1.eq2} \tag{2} \end{align} for any $k\in\mathbb{N}$, $\ell\in\mathbb{N}$, $u\in\wedge^{k}V$ and $v\in\wedge^{\ell}V$.

Proposition 2a. Let $f:\mathbb{K}^{n}\rightarrow\mathbb{K}^{n}$ be a $\mathbb{K}$-linear map. The map $\wedge^{n}f:\wedge^{n}\left( \mathbb{K} ^{n}\right) \rightarrow\wedge^{n}\left( \mathbb{K}^{n}\right) $ is multiplication by $\det f$. In other words, every $z\in\wedge^{n}\left( \mathbb{K}^{n}\right) $ satisfies \begin{align} \left( \wedge^{n}f\right) \left( z\right) =\left( \det f\right) z . \label{darij1.eq3} \tag{3} \end{align}

Let $\left( e_{1},e_{2},\ldots,e_{n}\right) $ be the standard basis of the $\mathbb{K}$-module $\mathbb{K}^{n}$. (Thus, $e_i$ is the column vector whose $i$-th entry is $1$ and whose all other entries are $0$.)

For every subset $K$ of $\left[ n\right] $, we define $e_K\in \wedge^{\left| K\right| }\left( \mathbb{K}^{n}\right) $ to be the element $e_{k_{1}}\wedge e_{k_{2}}\wedge\cdots\wedge e_{k_{\left| K\right| }}$, where $K$ is written in the form $K=\left\{ k_{1} <k_{2}<\cdots<k_{\left| K\right| }\right\} $.

It is well-known that, for every $k\in\mathbb{N}$, the family $\left( e_K\right) _{K\in\mathcal{P}_{k}\left( \left[ n\right] \right) }$ is a basis of the $\mathbb{K}$-module $\wedge^{k}\left( \mathbb{K}^{n}\right) $. Applying this to $k=n$, we conclude that the family $\left( e_K\right) _{K\in\mathcal{P}_{n}\left( \left[ n\right] \right) }$ is a basis of the $\mathbb{K}$-module $\wedge^{n}\left( \mathbb{K}^{n}\right) $. Since this family $\left( e_K\right) _{K\in\mathcal{P}_{n}\left( \left[ n\right] \right) }$ is the one-element family $\left( e_{\left[ n\right] }\right) $ (because the only $K\in\mathcal{P}_{n}\left( \left[ n\right] \right) $ is the set $\left[ n\right] $), this rewrites as follows: The one-element family $\left( e_{\left[ n\right] }\right) $ is a basis of the $\mathbb{K}$-module $\wedge^{n}\left( \mathbb{K}^{n}\right) $.

If $B$ is an $n\times n$-matrix and $k\in\mathbb{N}$, then evaluating the map $\wedge^{k}f_{B}$ on the elements of the basis $\left( e_K\right) _{K\in\mathcal{P}_{k}\left( \left[ n\right] \right) }$ of $\wedge ^{k}\left( \mathbb{K}^{n}\right) $, and expanding the results again in this basis gives rise to coefficients which are the $k\times k$-minors of $B$. More precisely:

Proposition 3. Let $B\in\mathbb{K}^{n\times n}$, $k\in\mathbb{N}$ and $J\in\mathcal{P}_{k}\left( \left[ n\right] \right) $. Then, \begin{align} \left( \wedge^{k}f_{B}\right) \left( e_{J}\right) = \sum\limits_{I\in\mathcal{P}_{k}\left( \left[ n\right] \right) }\det\left( B_{J} ^{I}\right) e_{I} . \end{align}

(This can be generalized: If $u\in\mathbb{N}$, $v \in \mathbb{N}$, $B\in\mathbb{K}^{u\times v}$, $k\in\mathbb{N}$ and $J\in\mathcal{P}_{k}\left( \left[ v\right] \right) $, then $\left( \wedge^{k}f_{B}\right) \left( e_{J}\right) = \sum\limits_{I\in\mathcal{P}_{k}\left( \left[ u\right] \right) }\det\left( B_{J} ^{I}\right) e_{I}$, where the elements $e_{J}\in\wedge^{k}\left( \mathbb{K}^{v}\right) $ and $e_{I}\in\wedge^{k}\left( \mathbb{K}^{u}\right) $ are defined as before but with $v$ and $u$ instead of $n$.)

Lemma 4. Let $K$ be a subset of $\left[ n\right] $. Then, \begin{align} e_K e_{\widetilde{K}}=\left( -1\right) ^{\sum K-\left( 1+2+\cdots+\left| K\right| \right) }e_{\left[ n\right] } . \end{align}

Proof of Lemma 4. Let $k = \left|K\right|$. Let $\sigma$ be the permutation $w\left( K\right) \in S_n$ defined above. Its first $k$ values $\sigma\left( 1\right) ,\sigma\left( 2\right) ,\ldots,\sigma\left( k\right) $ are the elements of $K$ in increasing order; thus, $e_{\sigma\left( 1\right) }\wedge e_{\sigma\left( 2\right) }\wedge\cdots\wedge e_{\sigma\left( k\right) }=e_K$. Its next $n-k$ values $\sigma\left( k+1\right) ,\sigma\left( k+2\right) ,\ldots,\sigma\left( n\right) $ are the elements of $\widetilde{K}$ in increasing order; thus, $e_{\sigma\left( k+1\right) }\wedge e_{\sigma\left( k+2\right) }\wedge\cdots\wedge e_{\sigma\left( n\right) }=e_{\widetilde{K}}$.

From $\sigma=w\left( K\right) $, we obtain $\left( -1\right) ^{\sigma }=\left( -1\right) ^{w\left( K\right) }=\left( -1\right) ^{\sum K-\left( 1+2+\cdots+\left| K\right| \right) }$ (by Lemma 2).

Now, it is well-known that \begin{align} e_{\sigma\left( 1\right) }\wedge e_{\sigma\left( 2\right) }\wedge \cdots\wedge e_{\sigma\left( n\right) }=\left( -1\right) ^{\sigma }\underbrace{e_{1}\wedge e_{2}\wedge\cdots\wedge e_{n}}_{=e_{\left[ n\right] }}=\left( -1\right) ^{\sigma}e_{\left[ n\right] } . \end{align} Hence, \begin{align} \left( -1\right) ^{\sigma}e_{\left[ n\right] } & = e_{\sigma\left( 1\right) }\wedge e_{\sigma\left( 2\right) }\wedge\cdots\wedge e_{\sigma\left( n\right) } \\ & = \underbrace{\left( e_{\sigma\left( 1\right) }\wedge e_{\sigma\left( 2\right) }\wedge\cdots\wedge e_{\sigma\left( k\right) }\right) }_{=e_K }\underbrace{\left( e_{\sigma\left( k+1\right) }\wedge e_{\sigma\left( k+2\right) }\wedge\cdots\wedge e_{\sigma\left( n\right) }\right) }_{=e_{\widetilde{K}}} \\ & = e_K e_{\widetilde{K}} . \end{align} Since $\left( -1\right) ^{\sigma}=\left( -1\right) ^{\sum K-\left( 1+2+\cdots+\left| K\right| \right) }$, this rewrites as $\left( -1\right) ^{\sum K-\left( 1+2+\cdots+\left| K\right| \right) }e_{\left[ n\right] }= e_K e_{\widetilde{K}}$. This proves Lemma 4. $\blacksquare$

Corollary 5. Let $B\in\mathbb{K}^{n\times n}$, $k\in\mathbb{N}$ and $J\in\mathcal{P}_{k}\left( \left[ n\right] \right) $. Then, every $K\in\mathcal{P}_{k}\left( \left[ n\right] \right) $ satisfies \begin{align} \left( \wedge^{k}f_{B}\right) \left( e_{J}\right) e_{\widetilde{K} }=\left( -1\right) ^{\sum K-\left( 1+2+\cdots+k\right) }\det\left( B_{J}^{K}\right) e_{\left[ n\right] } . \end{align}

Now, forget that we fixed $I$. We thus have proven that \begin{align} e_{I}e_{\widetilde{K}}=0 \text{ for every } I\in\mathcal{P}_{k}\left( \left[ n\right] \right) \text{ satisfying } I\neq K . \end{align} Hence, \begin{align} \sum\limits_{\substack{I\in\mathcal{P}_{k}\left( \left[ n\right] \right) ;\\I\neq K}}\det\left( B_{J}^{I}\right) \underbrace{e_{I}e_{\widetilde{K}} }_{=0}=0 . \label{darij1.eq4} \tag{4} \end{align} Proposition 3 yields \begin{align} \left( \wedge^{k}f_{B}\right) \left( e_{J}\right) =\sum\limits_{I\in \mathcal{P}_{k}\left( \left[ n\right] \right) }\det\left( B_{J} ^{I}\right) e_{I} . \end{align} Multiplying both sides of this equality by $e_{\widetilde{K}}$ from the right, we find \begin{align} & \left( \wedge^{k}f_{B}\right) \left( e_{J}\right) e_{\widetilde{K}} =\sum\limits_{I\in\mathcal{P}_{k}\left( \left[ n\right] \right) }\det\left( B_{J}^{I}\right) e_{I}e_{\widetilde{K}} \\ & = \det\left( B_{J}^{K}\right) e_K e_{\widetilde{K}}+\sum\limits_{\substack{I\in \mathcal{P}_{k}\left( \left[ n\right] \right) ;\\I\neq K}}\det\left( B_{J}^{I}\right) e_{I}e_{\widetilde{K}} \\ & = \det\left( B_{J}^{K}\right) \underbrace{e_K e_{\widetilde{K}} }_{\substack{=\left( -1\right) ^{\sum K-\left( 1+2+\cdots+\left| K\right| \right) }e_{\left[ n\right] }\\\text{(by Lemma 4)}}} \qquad \text{(by \eqref{darij1.eq4})} \\ & = \left( -1\right) ^{\sum K-\left( 1+2+\cdots+\left| K\right| \right) }\det\left( B_{J}^{K}\right) e_{\left[ n\right] } \\ & = \left( -1\right) ^{\sum K-\left( 1+2+\cdots+k\right) }\det\left( B_{J}^{K}\right) e_{\left[ n\right] } \end{align} (since $\left| K\right| =k$). This proves Corollary 5. $\blacksquare$

Proof of Theorem 1. Set $k=\left| I\right| =\left| J\right| $. Notice that $\left| \widetilde{I}\right| =n-k$ (since $\left| I\right| =k$) and $\left| \widetilde{J}\right| =n-k$ (similarly).

The maps $f_{A}$ and $f_{A^{-1}}$ are mutually inverse (since the map $\mathbb{K}^{n\times n}\rightarrow\operatorname*{End}\left( \mathbb{K} ^{n}\right) ,\ B\mapsto f_{B}$ is a ring homomorphism). Hence, the maps $\wedge^{n-k}f_{A}$ and $\wedge^{n-k}f_{A^{-1}}$ are mutually inverse (since $\wedge^{n-k}$ is a functor). Thus, $\left( \wedge^{n-k}f_{A}\right) \circ\left( \wedge^{n-k}f_{A^{-1}}\right) =\operatorname*{id}$. Now, $y=\left( \wedge^{n-k}f_{A^{-1}}\right) \left( e_{\widetilde{I}}\right) $, so that \begin{align} \left( \wedge^{n-k}f_{A}\right) \left( y\right) =\left( \wedge ^{n-k}f_{A}\right) \left( \left( \wedge^{n-k}f_{A^{-1}}\right) \left( e_{\widetilde{I}}\right) \right) =\underbrace{\left( \left( \wedge ^{n-k}f_{A}\right) \circ\left( \wedge^{n-k}f_{A^{-1}}\right) \right) }_{=\operatorname*{id}}\left( e_{\widetilde{I}}\right) =e_{\widetilde{I}} . \end{align} But \eqref{darij1.eq1} (applied to $V=\mathbb{K}^{n}$, $W=\mathbb{K}^{n}$, $f=f_{A}$, $\ell=n-k$, $a=e_{J}$ and $b=y$) yields \begin{align} \left( \wedge^{k}f_{A}\right) \left( e_{J}\right) \cdot\left( \wedge^{n-k}f_{A}\right) \left( y\right) =\left( \wedge^{n}f_{A}\right) \left( e_{J}y\right) . \end{align} Thus, \begin{align} & \left( \wedge^{n}f_{A}\right) \left( e_{J}y\right) =\left( \wedge ^{k}f_{A}\right) \left( e_{J}\right) \cdot\underbrace{\left( \wedge ^{n-k}f_{A}\right) \left( y\right) }_{=e_{\widetilde{I}}} \\ & =\left( \wedge^{k}f_{A}\right) \left( e_{J}\right) e_{\widetilde{I}} = \left( -1\right) ^{\sum I-\left( 1+2+\cdots+k\right) }\det\left( A_J^I \right) e_{\left[ n\right] } \end{align} (by Corollary 5, applied to $B=A$ and $K=I$).

Hence, \begin{align} & \left( -1\right) ^{\sum I-\left( 1+2+\cdots+k\right) }\det\left( A_J^I \right) e_{\left[ n\right] } \\ & = \left( \wedge^{n}f_{A}\right) \left( e_{J}y\right) =\underbrace{\left( \det f_{A}\right) }_{=\det A}e_{J}y \\ & \qquad \text{(by \eqref{darij1.eq3}, applied to $f=f_{A}$ and $z=e_{J}y$)} \\ & = \left( \det A\right) e_{J}y . \label{darij1.eq5} \tag{5} \end{align} But \eqref{darij1.eq2} (applied to $\ell=n-k$, $u=e_{J}$ and $v=y$) yields \begin{align} & e_{J} y = \left(-1\right)^{k \left(n-k\right)} \underbrace{y}_{=\left( \wedge^{n-k}f_{A^{-1}}\right) \left( e_{\widetilde{I}}\right) } \underbrace{e_{J}} _{=e_{\widetilde{\widetilde{J}}}} \\ & =\left( -1\right) ^{k\left( n-k\right) }\underbrace{\left( \wedge ^{n-k}f_{A^{-1}}\right) \left( e_{\widetilde{I}}\right) e_{\widetilde{\widetilde{J}}}}_{\substack{=\left( -1\right) ^{\sum \widetilde{J}-\left( 1+2+\cdots+\left( n-k\right) \right) }\det\left( \left( A^{-1}\right) _{\widetilde{I}}^{\widetilde{J}}\right) e_{\left[ n\right] }\\\text{(by Corollary 5, applied to }A^{-1}\text{, }n-k\text{, }\widetilde{I}\text{ and }\widetilde{J}\\\text{instead of }B\text{, }k\text{, }J\text{ and }K\text{)}}} \\ & =\left( -1\right) ^{k\left( n-k\right) }\left( -1\right) ^{\sum \widetilde{J}-\left( 1+2+\cdots+\left( n-k\right) \right) }\det\left( \left( A^{-1}\right) _{\widetilde{I}}^{\widetilde{J}}\right) e_{\left[ n\right] } \\ & =\left( -1\right) ^{k\left( n-k\right) +\sum\widetilde{J}-\left( 1+2+\cdots+\left( n-k\right) \right) }\det\left( \left( A^{-1}\right) _{\widetilde{I}}^{\widetilde{J}}\right) e_{\left[ n\right] } . \label{darij1.eq6} \tag{6} \end{align} But \begin{align} & k\left( n-k\right) +\underbrace{\sum\widetilde{J}}_{=\sum\left\{ 1,2,\ldots,n\right\} -\sum J}-\underbrace{\left( 1+2+\cdots+\left( n-k\right) \right) }_{=\sum\left\{ 1,2,\ldots,n-k\right\} } \\ & = k\left( n-k\right) +\sum\left\{ 1,2,\ldots,n\right\} -\sum J-\sum\left\{ 1,2,\ldots,n-k\right\} \\ & = \underbrace{\sum\left\{ 1,2,\ldots,n\right\} -\sum\left\{ 1,2,\ldots ,n-k\right\} }_{\substack{=\sum\left\{ n-k+1,n-k+2,\ldots,n\right\} \\=k\left( n-k\right) +\sum\left\{ 1,2,\ldots,k\right\} \\=k\left( n-k\right) +\left( 1+2+\cdots+k\right) }}+k\left( n-k\right) -\sum J \\ & = 2k\left( n-k\right) +\left( 1+2+\cdots+k\right) -\sum J \\ & \equiv-\left( 1+2+\cdots+k\right) -\sum J \mod 2 . \end{align} Hence, \begin{align} \left( -1\right) ^{k\left( n-k\right) +\sum\widetilde{J}-\left( 1+2+\cdots+\left( n-k\right) \right) }=\left( -1\right) ^{-\left( 1+2+\cdots+k\right) -\sum J} . \end{align} Thus, \eqref{darij1.eq6} rewrites as \begin{align} e_{J}y=\left( -1\right) ^{-\left( 1+2+\cdots+k\right) -\sum J}\det\left( \left( A^{-1}\right) _{\widetilde{I}}^{\widetilde{J}}\right) e_{\left[ n\right] } . \end{align} Hence, \eqref{darij1.eq5} rewrites as \begin{align} & \left( -1\right) ^{\sum I-\left( 1+2+\cdots+k\right) }\det\left( A_J^I \right) e_{\left[ n\right] } \\ & = \left( \det A\right) \left( -1\right) ^{-\left( 1+2+\cdots+k\right) -\sum J}\det\left( \left( A^{-1}\right) _{\widetilde{I}}^{\widetilde{J} }\right) e_{\left[ n\right] } . \end{align} We can "cancel" $e_{\left[ n\right] }$ from this equality (because if $\lambda$ and $\mu$ are two elements of $\mathbb{K}$ satisfying $\lambda e_{\left[ n\right] }=\mu e_{\left[ n\right] }$, then $\lambda=\mu$), and thus obtain \begin{align} & \left( -1\right) ^{\sum I-\left( 1+2+\cdots+k\right) }\det\left( A_J^I \right) \\ & = \left( \det A\right) \left( -1\right) ^{-\left( 1+2+\cdots+k\right) -\sum J}\det\left( \left( A^{-1}\right) _{\widetilde{I}}^{\widetilde{J} }\right) . \end{align} Dividing this equality by $\left( -1\right) ^{\sum I-\left( 1+2+\cdots +k\right) }$, we obtain \begin{align} & \det\left( A_J^I \right) \\ & = \left(\det A\right) \dfrac{\left( -1\right) ^{-\left( 1+2+\cdots+k\right) -\sum J}}{\left( -1\right) ^{\sum I-\left( 1+2+\cdots+k\right) }}\det\left( \left( A^{-1}\right) _{\widetilde{I}}^{\widetilde{J}}\right) \\ & = \left( -1\right) ^{\sum I+\sum J}\det A\cdot\det\left( \left( A^{-1}\right) _{\widetilde{I}}^{\widetilde{J}}\right) . \end{align} This proves Theorem 1. $\blacksquare$

• Theorem 1 is Lemma A.1 (e) in Sergio Caracciolo, Alan D. Sokal, Andrea Sportiello, Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians, arXiv:1105.6270v2 (published in: Advances in Applied Mathematics 50, 474--594 (2013)). In the paragraph following Theorem A.16, a proof is given using what the authors call "Grassmann-Berezin integration" (despite its name, a purely algebraic mock-calculus on the exterior algebra of a vector space).

• Theorem 1 is (1) in Pierre Lalonde, A non-commutative version of Jacobi's equality on the cofactors of a matrix, Discrete Mathematics 158 (1996), pp. 161--172. The goal of the paper is to generalize it to a (mildly) noncommutative setting.

• Theorem 1 is Additional exercise 19 in my Notes on the combinatorial fundamentals of algebra, version of 4 August 2016. The proof I give is fairly classical and similar to the ones given by Federico Poloni and Denis Serre, but requires no WLOG assumptions (instead of using the Schur complement, I use a cheap generalization of it which is Exercise 50 in the notes). The formal bookkeeping that leads to the sign $\left( -1\right) ^{\sum I+\sum J}$ takes a huge lot of space, although it is so easy to convince yourself of it with some handwaving that you might not notice that it requires any proof at all.

Note that every source uses different notations. What I call $A_{J}^{I}$ above is called $A_{IJ}$ in the paper by Caracciolo, Sokal and Sportiello, is called $A\left[ I,J\right] $ in Lalonde's paper, and is called $\operatorname*{sub}\nolimits_{w\left( I\right) }^{w\left( J\right) }A$ in my notes. Also, the $I$ and $J$ in the paper by Caracciolo, Sokal and Sportiello correspond to the $\widetilde{I}$ and $\widetilde{J}$ in Theorem 1 above.

Proof of Corollary 5. Let $I\in\mathcal{P}_{k}\left( \left[ n\right] \right) $ be such that $I\neq K$. Then, $I\not \subseteq K$ (since the sets $I$ and $K$ have the same size $k$). Hence, there exists some $z\in I$ such that $z\notin K$. Consider this $z$. We have $z\in I$ and $z\in\widetilde{K}$ (since $z\notin K$). Hence, both $e_{I}$ and $e_{\widetilde{K}}$ are "wedge products" containing the factor $e_{z}$; therefore, the product $e_{I} e_{\widetilde{K}}$ is a "wedge product" containing this factor twice. Thus, $e_{I}e_{\widetilde{K}}=0$.

But (1) (applied to $V=\mathbb{K}^{n}$, $W=\mathbb{K}^{n}$, $f=f_{A}$, $\ell=n-k$, $a=e_{J}$ and $b=\left( \wedge^{n-k}f_{A}\right) \left( y\right) $) yields

• Theorem 1 is Lemma A.1 (e) in Sergio Caracciolo, Alan D. Sokal, Andrea Sportiello, Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians, arXiv:1105.6270v2 (published in: Advances in Applied Mathematics 50, 474--594 (2013)). In the paragraph following Theorem A.16, a proof is given using what the authors call "Grassmann-Berezin integration" (despite its name, a purely algebraic mock-calculus on the exterior algebra of a vector space).

• Theorem 1 is (1) in Pierre Lalonde, A non-commutative version of Jacobi's equality on the cofactors of a matrix, Discrete Mathematics 158 (1996), pp. 161--172. The goal of the paper is to generalize it to a (mildly) noncommutative setting.

• Theorem 1 is Additional exercise 19 in my Notes on the combinatorial fundamentals of algebra, version of 12 December 2016. The first proof I give is fairly classical and similar to the ones given by Federico Poloni and Denis Serre, but requires no WLOG assumptions (instead of using the Schur complement, I use a cheap generalization of it which is Exercise 50 in the notes). The formal bookkeeping that leads to the sign $\left( -1\right) ^{\sum I+\sum J}$ takes a huge lot of space, although it is so easy to convince yourself of it with some handwaving that you might not notice that it requires any proof at all. The second proof is an expansion of an argument briefly outlined in D. Laksov, A. Lascoux, P. Pragacz, and A. Thorup, The LLPT Notes (Chapter SCHUR, proof of (1.9)).

Note that every source uses different notations. What I call $A_{J}^{I}$ above is called $A_{IJ}$ in the paper by Caracciolo, Sokal and Sportiello, is called $A\left[ I,J\right] $ in Lalonde's paper, and is called $\operatorname*{sub}\nolimits_{w\left( I\right) }^{w\left( J\right) }A$ in my notes. Also, the $I$ and $J$ in the paper by Caracciolo, Sokal and Sportiello correspond to the $\widetilde{I}$ and $\widetilde{J}$ in Theorem 1 above.

Proof of Corollary 5. Let $K \in \mathcal{P}_{k}\left( \left[ n\right] \right) $. Let $I\in\mathcal{P}_{k}\left( \left[ n\right] \right) $ be such that $I\neq K$. Then, $I\not \subseteq K$ (since the sets $I$ and $K$ have the same size $k$). Hence, there exists some $z\in I$ such that $z\notin K$. Consider this $z$. We have $z\in I$ and $z\in\widetilde{K}$ (since $z\notin K$). Hence, both $e_{I}$ and $e_{\widetilde{K}}$ are "wedge products" containing the factor $e_{z}$; therefore, the product $e_{I} e_{\widetilde{K}}$ is a "wedge product" containing this factor twice. Thus, $e_{I}e_{\widetilde{K}}=0$.

But (1) (applied to $V=\mathbb{K}^{n}$, $W=\mathbb{K}^{n}$, $f=f_{A}$, $\ell=n-k$, $a=e_{J}$ and $b=y$) yields

But

$e_{J}\underbrace{y}_{=\left( \wedge^{n-k}f_{A^{-1}}\right) \left( e_{\widetilde{I}}\right) }=e_{J}\left( \wedge^{n-k}f_{A^{-1}}\right) \left( e_{\widetilde{I}}\right) $

$=\left( -1\right) ^{k\left( n-k\right) }\left( \wedge^{n-k}f_{A^{-1} }\right) \left( e_{\widetilde{I}}\right) \underbrace{e_{J}} _{=e_{\widetilde{\widetilde{J}}}}$

(by (2), applied to $\ell=n-k$, $u=e_{J}$ and $v=\left( \wedge ^{n-k}f_{A^{-1}}\right) \left( e_{\widetilde{I}}\right) $)

$=\left( -1\right) ^{k\left( n-k\right) }\left( -1\right) ^{\sum \widetilde{J}-\left( 1+2+\cdots+\left( n-k\right) \right) }\det\left( \left( A^{-1}\right) _{\widetilde{I}}^{\widetilde{J}}\right) e_{\left[ n\right] }$.

(6) $=\left( -1\right) ^{k\left( n-k\right) +\sum\widetilde{J}-\left( 1+2+\cdots+\left\vert n-k\right\vert \right) }\det\left( \left( A^{-1}\right) _{\widetilde{I}}^{\widetilde{J}}\right) e_{\left[ n\right] }$.

$\left( -1\right) ^{k\left( n-k\right) +\sum\widetilde{J}-\left( 1+2+\cdots+\left\vert n-k\right\vert \right) }=\left( -1\right) ^{-\left( 1+2+\cdots+k\right) -\sum J}$.

But (2) (applied to $\ell=n-k$, $u=e_{J}$ and $v=y$) yields

$e_{J} y = \left(-1\right)^{k \left(n-k\right)} \underbrace{y}_{=\left( \wedge^{n-k}f_{A^{-1}}\right) \left( e_{\widetilde{I}}\right) } \underbrace{e_{J}} _{=e_{\widetilde{\widetilde{J}}}}$

$=\left( -1\right) ^{k\left( n-k\right) }\left( -1\right) ^{\sum \widetilde{J}-\left( 1+2+\cdots+\left( n-k\right) \right) }\det\left( \left( A^{-1}\right) _{\widetilde{I}}^{\widetilde{J}}\right) e_{\left[ n\right] }$

(6) $=\left( -1\right) ^{k\left( n-k\right) +\sum\widetilde{J}-\left( 1+2+\cdots+\left( n-k\right) \right) }\det\left( \left( A^{-1}\right) _{\widetilde{I}}^{\widetilde{J}}\right) e_{\left[ n\right] }$.

$\left( -1\right) ^{k\left( n-k\right) +\sum\widetilde{J}-\left( 1+2+\cdots+\left( n-k\right) \right) }=\left( -1\right) ^{-\left( 1+2+\cdots+k\right) -\sum J}$.