行列式
が成り立っている。例えば、線型性によって
| λ a 11 + μ a 11 ′ a 12 a 13 λ a 21 + μ a 21 ′ a 22 a 23 λ a 31 + μ a 31 ′ a 32 a 33 | = λ | a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 | + μ | a 11 ′ a 12 a 13 a 21 ′ a 22 a 23 a 31 ′ a 32 a 33 | {\displaystyle {\begin{vmatrix}\lambda a_{11}+\mu a_{11}'&a_{12}&a_{13}\\\lambda a_{21}+\mu a_{21}'&a_{22}&a_{23}\\\lambda a_{31}+\mu a_{31}'&a_{32}&a_{33}\end{vmatrix}}\,=\,\lambda \,{\begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{vmatrix}}\,{}+{}\,\mu \,{\begin{vmatrix}a_{11}'&a_{12}&a_{13}\\a_{21}'&a_{22}&a_{23}\\a_{31}'&a_{32}&a_{33}\end{vmatrix}}}