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MAT240 – December 1st
Basic Properties of det: Mnxn→F: 0 det(I) = 1
1. d e t ( E i , j ′ A ) = − d e t ( A ) ; | E i , j ′ | = − 1. [ N o t e : d e t ( E A ) = | E | | A | ] {\displaystyle det(E'_{i,j\,\!}A)=-det(A);|E'_{i,j\,\!}|=-1.[Note:det(EA)=|E||A|]}
2. d e t ( E i , c 2 A ) = d e t ( A ) ; | ( E i , j , c 2 | = 1 {\displaystyle det(E_{i,c\,\!}^{2}A)=det(A);|(E_{i,j,c\,\!}^{2}|=1}
3. d e t ( ( E i , j , c A ) = d e t ( A ) ; | ( E i , j , c 3 | = 1 ∗ A d d i n g a m u l t i p l e o f o n e r o w t o a n o t h e r d o e s n o t c h a n g e t h e d e t e r m i n a n t . T h e d e t e r m i n a n t o f a n y m a t r i x c a n b e c a l c u l a t e d u s i n g t h e p r o p e r t i e s a b o v e . {\displaystyle det((E_{i,j,c\,\!}A)=det(A);|(E_{i,j,c\,\!}^{3}|=1*Addingamultipleofonerowtoanotherdoesnotchangethedeterminant.Thedeterminantofanymatrixcanbecalculatedusingthepropertiesabove.}
![{\displaystyle det(E'_{i,j\,\!}A)=-det(A);|E'_{i,j\,\!}|=-1.[Note:det(EA)=|E||A|]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93b201c57138463bb97bc0ea059970a95c7d0fd0)
