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Hi, I'm having trouble getting started on the following question and would like some help.

Q. Let U be an open subset of R^n and V be an open subset of R^m. Assume that [itex]f:U \to V[/itex] is a C^1 function with a C^1 inverse [itex]g:V \to U[/itex]. (Thus (f o g) is the identity function [itex]1_V :V \to V[/itex] and (g o f) is the identity function [itex]1_U :U \to U[/itex].)

a) Show that the matrix [itex]Df\left( {\mathop {x_0 }\limits^ \to } \right)[/itex] is invertible with inverse [itex]Dg\left( {f\left( {\mathop {x_0 }\limits^ \to } \right)} \right)[/itex] for each x_0 in U.

I'm thinking that I might need to use matrix multiplication for example AA^-1 = I or something like that. I know that the derivative matrices for f and g evaluated at x_0 are defined since they are both C^1. Those are just some rough ideas and I don't know where to begin. Is there something obvious that I should I work with to get started? Any help would be good thanks.

Q. Let U be an open subset of R^n and V be an open subset of R^m. Assume that [itex]f:U \to V[/itex] is a C^1 function with a C^1 inverse [itex]g:V \to U[/itex]. (Thus (f o g) is the identity function [itex]1_V :V \to V[/itex] and (g o f) is the identity function [itex]1_U :U \to U[/itex].)

a) Show that the matrix [itex]Df\left( {\mathop {x_0 }\limits^ \to } \right)[/itex] is invertible with inverse [itex]Dg\left( {f\left( {\mathop {x_0 }\limits^ \to } \right)} \right)[/itex] for each x_0 in U.

I'm thinking that I might need to use matrix multiplication for example AA^-1 = I or something like that. I know that the derivative matrices for f and g evaluated at x_0 are defined since they are both C^1. Those are just some rough ideas and I don't know where to begin. Is there something obvious that I should I work with to get started? Any help would be good thanks.

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