The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 10.14.6. Let $R$ be a ring. Let $n \geq m$. Let $A = (a_{ij})$ be an $n \times m$ matrix with coefficients in $R$, written in block form as

\[ A = \left( \begin{matrix} A_1 \\ A_2 \end{matrix} \right) \]

where $A_1$ has size $m \times m$. Let $B$ be the adjugate (transpose of cofactor) matrix to $A_1$. Then

\[ AB = \left( \begin{matrix} f 1_{m \times m} \\ C \end{matrix} \right) \]

where $f = \det (A_1)$ and $c_{ij}$ is (up to sign) the determinant of the $m \times m$ minor of $A$ corresponding to the rows $1, \ldots , \hat j, \ldots , m, i$.

Proof. Since the adjugate has the property $A_1B = B A_1 = f$ the first block of the expression for $AB$ is correct. Note that

\[ c_{ij} = \sum \nolimits _ k a_{ik}b_{kj} = \sum (-1)^{j + k}a_{ik} \det (A_1^{jk}) \]

where $A_1^{ij}$ means $A_1$ with the $j$th row and $k$th column removed. This last expression is the row expansion of the determinant of the matrix in the statement of the lemma. $\square$


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