Select mxm matrix from mxn matrix and solve systems of linear equations

Hi everyone, 

I has started using Matlab recently. There are two problems in my following code:
+ Firstly, I would like to choose different sets of m columns from n columns matrix. e.g,5 columns from 10 columns. 
There must be 252 sets, however, my code randomly select so there are some sets are the same (only the position of the columns are changed).

+ Secondly, when solving the systems of linear equations with matrix: a=xb then x=a\b, there appears error "Warning: Matrix is close to singular 
or badly scaled.Results may be inaccurate" since the det(a)~0. I have set "if (det(a)<=-1e-12) | (det(a)>=1e-12)" but it seems not so good.

Hope to receive your help. Thank you so much in advance.


A =[ 0     0     2     2     6     3     4     5     2     1
     2     2     0     0     0     1     0     0     0     1
     0     2     0     1     2     0     1     0     0     1
     0     0     0     0     0     0     0     1     1     0
     3     7     1     3     7     3     4     4     3     4 ];
 %choose different sets of 5 columns from 10 columns: 10!/(5!5!)=252 sets
for i=1:252
    c=randperm(length(A),5); 
    a=A(:,c);
b=[5.8; 2.4; 1.6; 0.2; 10];
%Error "Warning: Matrix is close to singular or badly scaled" 
%if det(a)~=0
if (det(a)<=-1e-12) | (det(a)>=1e-12)
   x=a\b;
%Find non-negative and non-Infinities x values
    if (x(:,:)>=0) & (x(:,:)~=Inf) 
        disp(x)
        disp(a)
    end
end
end

Answer 1

[J. Alex Lee]

Does this help with first problem: https://www.mathworks.com/help/matlab/ref/nchoosek.html

Second problem, maybe make tolerance more stringent?

[Thuy Tran]

Thanks a lot for your quick reply. However, "nchoosek" doesn't work for my case. I want to select 5 colums (5x5 matrix) from 10 columns (5x10 matrix). From my code, there are 252 output but some of them are the same (only different in column's position), e.g, [5 2; 3 1] and [2 5; 1 3] are the same.

For the secons problem, make tolerance more stringent is also the only way I can think of up to now.

[J. Alex Lee]

I'm not sure I understand your examples, which are 2x2...but it sounds like you are confirming that you wish to find ALL the unique combinations of K columns of a matrix that has N columns. In this case, nchoosek() should do exactly what you need

clc;
clear;

A =[ 0     0     2     2     6     3     4     5     2     1
	2     2     0     0     0     1     0     0     0     1
	0     2     0     1     2     0     1     0     0     1
	0     0     0     0     0     0     0     1     1     0
	3     7     1     3     7     3     4     4     3     4 ];

K = 5
N = size(A,2)

colsets = nchoosek(1:N,K);

b = [5.8; 2.4; 1.6; 0.2; 10];

% confirm that you have the right number of combinations
nCombinations = size(colsets,1)
for i = 1:nCombinations
	
	ASub = A(:,colsets(i,:));
	
	if rank(ASub) < K
		fprintf('skip combination %3d\n',i)
	else
		x = ASub \ b;
	end
end

[Thuy Tran]

Sorry, I misunderstood your previous sugesstion. Thank you so much for your help. It is exactly what I need.