行列式に対するライプニッツの明示公式
, a n ) {\displaystyle {\begin{aligned}F(\mathbf {a} ^{1},\dotsc ,\mathbf {a} ^{j}+\mathbf {b} ^{j},\dotsc ,\mathbf {a} ^{n})&=\sum _{\sigma \in S_{n}}\operatorname {sgn}(\sigma )(a_{\sigma (j)}^{j}+b_{\sigma (j)}^{j})\prod _{i=1,\dotsc ,n \atop i\neq j}a_{\sigma (i)}^{i}\\&=\sum _{\sigma \in S_{n}}\operatorname {sgn}(\sigma ){\biggl (}{\Bigl (}\prod _{i=1}^{n}a_{\sigma (i)}^{i}{\Bigr )}+{\Bigl (}b_{\sigma (j)}^{j}\prod _{i=1,\dotsc ,n \atop i\neq j}a_{\sigma (i)}^{i}{\Bigr )}{\biggr )}\\&={\Bigl (}\sum _{\sigma \in S_{n}}\operatorname {sgn}(\sigma )\prod _{i=1}^{n}a_{\sigma (i)}^{i}{\Bigr )}+{\Bigl (}\sum _{\sigma \in S_{n}}\operatorname {sgn}(\sigma )b_{\sigma (j)}^{j}\prod _{i=1,\dotsc ,n \atop i\neq j}a_{\sigma (i)}^{i}{\Bigr )}\\&=F(\mathbf {a} ^{1},\dotsc ,\mathbf {a} ^{j},\dotsc ,\mathbf {a} ^{n})+F(\mathbf {a} ^{1},\dotsc ,\mathbf {b} ^{j},\dotsc ,\mathbf {a} ^{n})\end{aligned}}}