Question Number 134008 by AbderrahimMaths last updated on 26/Feb/21
![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](Q134008.png)
$$\:\:\:\:{we}\:{consider}\:{that}\:{application}\:{n}\geqslant\mathrm{1} \\ $$$$\:\:{det}\::\:{M}_{{n}} \left(\mathbb{R}\right)\rightarrow\mathbb{R} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{A} {det}\left({A}\right) \\ $$$$\mathrm{1}−{verify}\:{that}\:\forall{H}\in{M}_{{n}} \left(\mathbb{R}\right)\:{and}\:{t}\in\mathbb{R} \\ $$$$\:{if}\:{A}={I}_{{n}} \Rightarrow{det}\left({A}+{tH}\right)=\mathrm{1}+{t}.{Tr}\left({H}\right)+\circ\left({t}\right) \\ $$$$\mathrm{2}−{suppose}\:{that}:\:{A}\in{GL}_{{n}} \left(\mathbb{R}\right) \\ $$$$\:{prouve}\:{that}\:{the}\:{differntial}\:{of}\:{det}\:{in}\:{A}\:{is}\:{given}\:{by}: \\ $$$$\:\:\:{H} {Tr}\left[\left({com}\left({A}\right)\right)^{{T}} {H}\right] \\ $$$$\mathrm{3}−{determinate}\:{the}\:{differential}\:{of}\:{determinant}\:{of}\:{a}\:{matrix}\:{in}\:{general}\:{case}. \\ $$$$\left({Use}\:{the}\:{density}\:{of}\:{GL}_{{n}} \left(\mathbb{R}\right)\:{in}\:{M}_{{n}} \left(\mathbb{R}\right)\right. \\ $$$$\:{Tr}:\:{trace}\:{of}\:{matrix} \\ $$$$\left({com}\left({A}\right)\right)^{{T}} :\:{transpose}\:{of}\:{the}\:{comatrix} \\ $$