we consider that application n≥1 det : M_n (R)→R A det(A) 1−verify that ∀H∈M_n (R) and t∈R if A=I_n ⇒det(A+tH)=1+t.Tr(H)+○(t) 2−suppose that: A∈GL_n (R) prouve that the differntial of det in A is given by: H Tr[(com(A))^T H] 3−determinate the differential of determinant of a matrix in general case. (Use the density of GL_n (R) in M_n (R) Tr: trace of matrix (com(A))^T : transpose of the comatrix


Question and Answers Forum

All Questions Topic List
None Questions
Previous in All Question Next in All Question
Previous in None Next in None
Question Number 134008 by AbderrahimMaths last updated on 26/Feb/21

$w e c o n s i d e r t h a t a p p l i c a t i o n n ⩾ 1$ $d e t : M_{n} (R) \to R$ $A d e t (A)$ $1 - v e r i f y t h a t \forall H \in M_{n} (R) a n d t \in R$ $i f A = I_{n} \Rightarrow d e t (A + t H) = 1 + t . T r (H) + \circ (t)$ $2 - s u p p o s e t h a t : A \in {G L}_{n} (R)$ $p r o u v e t h a t t h e d i f f e r n t i a l o f d e t i n A i s g i v e n b y :$ $H T r [{(c o m (A))}^{T} H]$ $3 - d e t e r m i n a t e t h e d i f f e r e n t i a l o f d e t e r m i n a n t o f a m a t r i x i n g e n e r a l c a s e .$ $(U s e t h e d e n s i t y o f {G L}_{n} (R) i n M_{n} (R)$ $T r : t r a c e o f m a t r i x$ ${(c o m (A))}^{T} : t r a n s p o s e o f t h e c o m a t r i x$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum