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If the inverse matrix ([tex]A^{-1}[/tex])is plotted in the original space, then does it have any relation to the original matrix([tex]A[/tex])?

What I mean by that is, if you have a function [tex]y=f(x)[/tex] in 2 D space, and you find the inverse function [tex]x=f^{-1}(y)[/tex] the inverse function is a reflection of the function [tex]y=f(x)[/tex] about the line [tex]y=x[/tex]. Does the inverse matrix ([tex]A^{-1}[/tex]) have any such relation to the original matrix([tex]A[/tex])?