\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
Let \(A \in \mathrm{GL}_n(\mathbb{K})\). Show that the map \(\Phi: M_{n,1}(\mathbb{K}) \to M_{n,1}(\mathbb{K})\) defined by \(\Phi(B) := X\), where \(X\) is the unique solution of \(A X = B\), is a bijection. Identify its inverse.
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