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→Критерий обратимости матрицы
==Критерий обратимости матрицыТеорема умножения определителей ==
{{Теорема
|statement=
Пусть <tex>\mathcal{A}</tex>, <tex>\mathcal{B} \colon X \to X</tex> (автоморфизм). <br> Тогда <tex>\det (\mathcal{A} \cdot \mathcal{B}) = \det \mathcal{A} \cdot \det \mathcal{B}</tex>
|proof =
<tex>\det (\mathcal{A} \cdot \mathcal{B}) {e_1} \land {e_2} \land... \land{e_n} = </tex><br><tex>(\mathcal{A} \cdot \mathcal{B})^{\wedge_n}{e_1} \land {e_2} \land... \land{e_n} = </tex><br><tex>(\mathcal{A} \cdot \mathcal{B}) {e_1} \land (\mathcal{A} \cdot \mathcal{B}) {e_2} \land ... \land (\mathcal{A} \cdot \mathcal{B}) {e_n} = </tex><br><tex>\mathcal{A} (\mathcal{B} {e_1}) \land \mathcal{A} (\mathcal{B} {e_2}) \land ... \land \mathcal{A} (\mathcal{B} {e_n}) =</tex><br><tex>\mathcal{A}^{\wedge_n}(\mathcal{B} {e_1} \land \mathcal{B} {e_2} \land ... \land \mathcal{B} {e_n})= </tex><br><tex>\det \mathcal{A} \cdot (\mathcal{B} {e_1} \land \mathcal{B} {e_2} \land ... \land \mathcal{B} {e_n}) = </tex><br><tex>\det \mathcal{A} \cdot \mathcal{B}^{\wedge_n}({e_1} \land {e_2} \land ... \land {e_n}) = </tex><br><tex>\det \mathcal{A} \cdot \det \mathcal{B} \cdot {e_1} \land {e_2} \land ... \land {e_n} </tex><br> т.е. <tex> \det (\mathcal{A} \cdot \mathcal{B}) {e_1} \land {e_2} \land... \land{e_n} = </tex><br><tex>
\det \mathcal{A} \cdot \det \mathcal{B} \cdot {e_1} \land {e_2} \land ... \land {e_n}
</tex>
}}
