데스나노-자코비 항등식
개요
- 행렬의 minor 사이에 성립하는 항등식
예
\(n=3\)인 경우
\[ \begin{align} \det \left( \begin{array}{ccc} a_{1,1} & a_{1,2} & a_{1,3} \\ a_{2,1} & a_{2,2} & a_{2,3} \\ a_{3,1} & a_{3,2} & a_{3,3} \\ \end{array} \right) \det\left( \begin{array}{c} a_{2,2} \\ \end{array} \right)\\ = \det \left( \begin{array}{cc} a_{1,1} & a_{1,2} \\ a_{2,1} & a_{2,2} \\ \end{array} \right)\det\left( \begin{array}{cc} a_{2,2} & a_{2,3} \\ a_{3,2} & a_{3,3} \\ \end{array} \right)-\det\left( \begin{array}{cc} a_{1,2} & a_{1,3} \\ a_{2,2} & a_{2,3} \\ \end{array} \right)\det\left( \begin{array}{cc} a_{2,1} & a_{2,2} \\ a_{3,1} & a_{3,2} \\ \end{array} \right) \end{align} \]
\(n=4\)인 경우
\[ \begin{align} \det\left( \begin{array}{cccc} a_{1,1} & a_{1,2} & a_{1,3} & a_{1,4} \\ a_{2,1} & a_{2,2} & a_{2,3} & a_{2,4} \\ a_{3,1} & a_{3,2} & a_{3,3} & a_{3,4} \\ a_{4,1} & a_{4,2} & a_{4,3} & a_{4,4} \\ \end{array} \right)\det \left( \begin{array}{cc} a_{2,2} & a_{2,3} \\ a_{3,2} & a_{3,3} \\ \end{array} \right) \\ = \det \left( \begin{array}{ccc} a_{1,1} & a_{1,2} & a_{1,3} \\ a_{2,1} & a_{2,2} & a_{2,3} \\ a_{3,1} & a_{3,2} & a_{3,3} \\ \end{array} \right)\det\left( \begin{array}{ccc} a_{2,2} & a_{2,3} & a_{2,4} \\ a_{3,2} & a_{3,3} & a_{3,4} \\ a_{4,2} & a_{4,3} & a_{4,4} \\ \end{array} \right)-\det\left( \begin{array}{ccc} a_{1,2} & a_{1,3} & a_{1,4} \\ a_{2,2} & a_{2,3} & a_{2,4} \\ a_{3,2} & a_{3,3} & a_{3,4} \\ \end{array} \right)\det\left( \begin{array}{ccc} a_{2,1} & a_{2,2} & a_{2,3} \\ a_{3,1} & a_{3,2} & a_{3,3} \\ a_{4,1} & a_{4,2} & a_{4,3} \\ \end{array} \right) \end{align} \]
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