please showw a rigorous proof of all parts for full rating thank you Let U : = {
ID: 2981109 • Letter: P
Question
please showw a rigorous proof of all parts for full rating thank you
Let U : = {A Rnxn : A is ail invertible matrix} . Show that U is an open subset ofRnxn. Hint. Use that A U if and only if det(A) 0. Prove that det is a continuous function on Rnxn. Recall that a function is continuous if and only if the preimages of open sets are open. Prove that the map f : UrightarrowU defined by f(A) : = A-1 is differentiate with d fA(B) = -A-1BA-1 (1) Hints. In order to prove that f is differentiate, you can use e. g. Cramer's rule to show that all entries of f have continuous partial derivatives in A. In order to obtain (1), it is easiest to compute the directional derivative df/dtf{A + tB)|t=o from the identity Af(A) = 1 for all A U.Explanation / Answer
det is a continuous function det(A) is non zero implies there exist another matrix B such that det (A+B) is non zero, so A+B is non singular so U is open as a function is continuous if and only if its preimage of open sets are open