What does the determinant of a matrix mean physically

Each  n×nn×n  matrix describes a linear transformation  T:Rn→Rn.T:Rn→Rn.  If you take a figure S⊆Rn,S⊆Rn,  then that transformation maps it to its image, another figure T(S)⊆Rn.T(S)⊆Rn.    In the case

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