How to find every minor determinant of a matrix?

Hello!

I am searching for a convenient way to calculate every minor determinant of a matrix. For example, given the matrix

    2.8722    1.7788    0.2750    0.3751    1.5872 
    0.9906    1.9001    0.5389    0.1804    0.3458
    0.3924    1.2647    2.1009    0.9696    0.3883
    0.4222    0.2286    0.3922    2.0427    1.3831
    1.5836    0.5931    0.1603    1.1563    3.2576

I want to calculate the determinand of every 2x2, 3x3 and 4x4 minor (5x5 is trivial). For this matrix the 3x3 will be, for example

    2.8722    1.7788    0.2750   
    0.9906    1.9001    0.5389 
    0.3924    1.2647    2.1009

    2.8722    1.7788    0.3751
    0.9906    1.9001    0.1804
    0.3924    1.2647    0.9696

    2.8722    1.7788    1.5872 
    0.9906    1.9001    0.3458
    0.3924    1.2647    0.3883

And so on...

Is there any quick way you can think about?

Many thanks!

Answer 1

[guy sharon]

I worked something out. Here is my take on this:

function [det_mat] = detMat(m)

len = length(m);
det_mat{1} = m;
det_mat{len} = det(m);

for order = 2 : len - 1
    perms = nchoosek(1:len,order);
    for i = 1 : length(perms)
        for j = 1 : length(perms)
            det_mat{order}(i,j) = det(m(perms(i,1:end),perms(j,1:end)));
        end
    end
end

Answer 2

[Andrew Warr]

Working from Guy Sharon's code start above, I've fixed the errors of Minor placement and matrices larger that 3 x 3 failing on the zero indexes.

I've tested it on matrices up to 5 x 5 and back checked against inv(Original)

This may be useful for a student checking their adjunct method working of a test writer wanting to test their marking guide.

function [det_mat] = detMat(m)
[nr, nc] = size (m);
if (nr == nc) 
  len = length(m);
  det_mat{1} = m;           %% Input matrix in first place
  det_mat{5} = det(m);  %% Determinant in fourth place
  for order = len - 1 : len - 1
     perms = nchoosek(1:len,order);      % Map to skip rows and columns
       for i = 1 : length(perms)
          for j = 1 : length(perms)
              %% The Minors
              det_mat{2}(len+1-i,len+1-j) = det(m(perms(i,1:end),perms(j,1:end)));
              %% The Masked Minors   
              if (mod(abs(i-j),2) == 1)
                det_mat{3}(len+1-i,len+1-j) = -1 * (det_mat{2}(len+1-i,len+1-j));
              else
                det_mat{3}(len+1-i,len+1-j) = det_mat{2}(len+1-i,len+1-j);
              endif
              temp_mat(len+1-i,len+1-j) = det_mat{3}(len+1-i,len+1-j);              
          end
       end
  end
  %% Make Adjunct by Transpose off-diagonal elements %% Yes, you can use adjoint(A)
  for j = 1 : length(perms)
    for i = 1 : length(perms)
       det_mat{4}(i,j) = temp_mat(j,i);
    end
  end
  %% The final Inverse Yes,you can use inv(m)
  det_mat{6} = 1/det_mat{5}*det_mat{4};
  det_mat{7} = [' 1)Your matrix ' ' 2)Minors ' ' 3)Masked ' ' 4)Adjunct ' ' 5)Determinant ' ' 6)Inverse'];
  
elseif  ((nr =! nc))              %% double bracket suppresses warning????
  disp("Error: Not a square matrix")
endif 

[Alireza Esmaily]

Hi dear Andrew warr

but there is a problem in this code

perms code has a imitation on length of matrices.

MATLAB Help:

perms(v) is practical when length(v) is less than about 10.