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For a Jacobian Matrix where the elements are represented as J11 J12 J13 J14 J21 J22 J23 J24 J31 J32 J33 J34 J41 J42 J43 J44

what is the inverse jacobian of this Matrix?

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  • 3
    $\begingroup$ Inverse[ArrayReshape[{J11,J12,J13,J14,J21,J22,J23,J24,J31,J32,J33,J34,J41,J42,J43,J44},{4,4}]] $\endgroup$ – flinty Sep 4 at 21:04
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"The Jacobian" often refers to the Jacobian determinant. For a $4\times4$ matrix

J = Partition[{J11, J12, J13, J14, J21, J22, J23, J24, J31, J32, J33, J34, J41, J42, J43, J44},
              4]

(*    {{J11, J12, J13, J14},
       {J21, J22, J23, J24},
       {J31, J32, J33, J34},
       {J41, J42, J43, J44}}    *)

this determinant is

Det[J]

(*    J14 J23 J32 J41 - J13 J24 J32 J41 - J14 J22 J33 J41 + 
      J12 J24 J33 J41 + J13 J22 J34 J41 - J12 J23 J34 J41 - 
      J14 J23 J31 J42 + J13 J24 J31 J42 + J14 J21 J33 J42 - 
      J11 J24 J33 J42 - J13 J21 J34 J42 + J11 J23 J34 J42 + 
      J14 J22 J31 J43 - J12 J24 J31 J43 - J14 J21 J32 J43 + 
      J11 J24 J32 J43 + J12 J21 J34 J43 - J11 J22 J34 J43 - 
      J13 J22 J31 J44 + J12 J23 J31 J44 + J13 J21 J32 J44 - 
      J11 J23 J32 J44 - J12 J21 J33 J44 + J11 J22 J33 J44      *)

The determinant of the inverse is equal to the inverse of the determinant:

Det[Inverse[J]] == 1/Det[J] // FullSimplify

(*    True    *)

1/Det[J]

(*    1/(J14 J23 J32 J41 - J13 J24 J32 J41 - J14 J22 J33 J41 + 
         J12 J24 J33 J41 + J13 J22 J34 J41 - J12 J23 J34 J41 - 
         J14 J23 J31 J42 + J13 J24 J31 J42 + J14 J21 J33 J42 - 
         J11 J24 J33 J42 - J13 J21 J34 J42 + J11 J23 J34 J42 + 
         J14 J22 J31 J43 - J12 J24 J31 J43 - J14 J21 J32 J43 + 
         J11 J24 J32 J43 + J12 J21 J34 J43 - J11 J22 J34 J43 - 
         J13 J22 J31 J44 + J12 J23 J31 J44 + J13 J21 J32 J44 - 
         J11 J23 J32 J44 - J12 J21 J33 J44 + J11 J22 J33 J44)     *)
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