I'm grading for a graduate real analysis course and looking through the text, trying to do some problems before the term starts so I can stay ahead of things. I found the following exercise in the section on the Inverse and Implicit Function Theorems.
Identify $\mathbb{R}^{2n}$ with $\mathbb{C}^n$ by $(x , y) \mapsto x + iy$. Let $U \subseteq \mathbb{R}^{2n}$ be open, and let $F : U \to \mathbb{R}^{2n}$ be $C^1$. Assume $p \in U, DF(p)$ invertible. If $F^{-1}: V \to U$ is the inverse to $F$, show that if $F$ is holomorphic, then so is $F^{-1}$.
The definition of holomorphic for a map $F : \mathbb{R}^{2} \to \mathbb{R}^{2}$ is given in an earlier exercise as satisfying the Riemann-Cauchy equations, or equivalently as $$DF(x) \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} DF(x),$$ where the exercise is to show that these are equivalent, then generalize to higher dimensions.
I'm at an impasse here with how to show this fact using real analysis. It seems fairly clear to me that if I could understand these as maps from $\mathbb{C}^{n}$ to $\mathbb{C}^{n}$, I could just say that the inverse of an invertible complex matrix is an invertible complex matrix and call it, but I don't know how to do this one with these techniques.
I figured that what I'd want to do is show that if $DF(x)$ satisfied the Cauchy-Riemann equations, then so did $DF(x)^{-1}$. I tried using some linear algebra trickery, describing the "Cauchy-Riemann matrices" as real $2n \times 2n$ matrices with blocks of the form $\begin{smallmatrix} a & -b \\ b & a \end{smallmatrix}$, and constructing an $\mathbb{R}$-algebra isomorphism between these and the $n \times n$ complex matrices so I could manipulate the complex matrices, but showing the isomorphism was multiplicative got out of hand and I felt fairly sure the author had an easier, more analytic approach in mind when he wrote it.
So please help me! I'm at a loss on how to show the inverse of a non-singular holomorphic map is holomorphic, and would love your advice. Thanks in advance!