Determinants, Matrix Norms, Inverse Mapping Theorem

The purpose of this notes is to present some useful facts about matrices and determinants and a proof of the inverse mapping theorem that is rather different from the one in Apostol. Notation: Mn(R) denotes the set of all n× n real matrices. Determinants: If A ∈ Mn(R), we can consider the rows of A: r1, . . . , rn. These are elements of R, considered as row vectors. Conversely, given n row vectors r1, . . . , rn, we can stack them up into an n× n matrix A. Thus we can think of a function f on matrix space Mn(R) as a function of n R-valued variables or vice versa: f(A)←→ f(r1, . . . , rn).