Question Number 133482 by AbderrahimMaths last updated on 22/Feb/21
$$\:\:\:\:{we}\:{consider}\:{that}\:{application}\:{n}\geqslant\mathrm{1} \\ $$$$\:\:{det}\::\:{M}_{{n}} \left(\mathbb{R}\right)\rightarrow\mathbb{R} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{A} {det}\left({A}\right) \\ $$$$\mathrm{1}−{verify}\:{that}\:\forall{H}\in{M}_{{n}} \left(\mathbb{R}\right)\:{and}\:{t}\in\mathbb{R} \\ $$$$\:{if}\:{A}={I}_{{n}} \Rightarrow{det}\left({A}+{tH}\right)=\mathrm{1}+{t}.{Tr}\left({H}\right)+\circ\left({t}\right) \\ $$$$\mathrm{2}−{suppose}\:{that}:\:{A}\in{GL}_{{n}} \left(\mathbb{R}\right) \\ $$$$\:{prouve}\:{that}\:{the}\:{differntial}\:{of}\:{det}\:{in}\:{A}\:{is}\:{given}\:{by}: \\ $$$$\:\:\:{H} {Tr}\left[\left({com}\left({A}\right)\right)^{{T}} {H}\right] \\ $$$$\:{Tr}:\:{trace}\:{of}\:{matrix} \\ $$$$\left({com}\left({A}\right)\right)^{{T}} :\:{transpose}\:{of}\:{the}\:{comatrix} \\ $$
Commented by AbderrahimMaths last updated on 25/Feb/21
$$\bullet{j} \\ $$