Check documentation for the function in question. (As a general rule, one should do that before posting.) The Details section explains what it does and in particular the ordering of the result.

Here is the example at hand (which should be in the body of the main message, not posted in a response). The code to get the different ordering is what is shown in the Details of the reference guide page.

mat3 = Array[a, {3, 3}];
Minors[mat3]
Reverse[Minors[mat3], {1, 2}]


(* Out[142]= {{-a[1, 2] a[2, 1] + a[1, 1] a[2, 2], -a[1, 3] a[2, 1] + 
   a[1, 1] a[2, 3], -a[1, 3] a[2, 2] + 
   a[1, 2] a[2, 3]}, {-a[1, 2] a[3, 1] + 
   a[1, 1] a[3, 2], -a[1, 3] a[3, 1] + 
   a[1, 1] a[3, 3], -a[1, 3] a[3, 2] + 
   a[1, 2] a[3, 3]}, {-a[2, 2] a[3, 1] + 
   a[2, 1] a[3, 2], -a[2, 3] a[3, 1] + 
   a[2, 1] a[3, 3], -a[2, 3] a[3, 2] + a[2, 2] a[3, 3]}}

Out[143]= {{-a[2, 3] a[3, 2] + a[2, 2] a[3, 3], -a[2, 3] a[3, 1] + 
   a[2, 1] a[3, 3], -a[2, 2] a[3, 1] + 
   a[2, 1] a[3, 2]}, {-a[1, 3] a[3, 2] + 
   a[1, 2] a[3, 3], -a[1, 3] a[3, 1] + 
   a[1, 1] a[3, 3], -a[1, 2] a[3, 1] + 
   a[1, 1] a[3, 2]}, {-a[1, 3] a[2, 2] + 
   a[1, 2] a[2, 3], -a[1, 3] a[2, 1] + 
   a[1, 1] a[2, 3], -a[1, 2] a[2, 1] + a[1, 1] a[2, 2]}} *)

Regarding the ordering, it is lexicographic based on rows and columns retained in constructing the given minor.

Minors will not be changing.

Below mentioned code is given for finding the Minors of a matrix:

mat5=Array[Subscript[a,## ] &,{3,3}]//MatrixForm

Minors[mat5]//MatrixForm

Instead, we need the output of the following command:

minorsMatrix[m_List?MatrixQ]:=Map[Reverse,Minors[m],{0,1}]
minorsMatrix[mat5]//MatrixForm