Totally Invertible Submatrices
Jelly, 26 24 23 20 19 17 16 bytes
-1 byte thanks to @miles (unnecessary for each, €, when taking determinants)
-2 bytes, @miles again! (unnecessary chain separation, and use of Ð€ quick)
ZœcLÆḊ
œcÐ€JÇ€€Ȧ

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How?
œcÐ€JÇ€€Ȧ  - Main link: matrix as an array, M
    J      - range of length -&gt; [1,2,...,len(a)] (n)
  Ð€       - for each of right argument
œc         -     combinations of M numbering n
     Ç€€   - call the last link (1) as a monad for €ach for €ach
        Ȧ  - all truthy (any determinant of zero results in 0, otherwise 1)
                 (this includes an implicit flattening of the list)

ZœcLÆḊ - Link 1, determinants of sub-matrices: row selection, s
Z      - transpose s
   L   - length of s
 œc    - combinations of transposed s numbering length of s
    ÆḊ - determinant