2.77 Determine if a set of linear equations A x= b has a solution and what type of solution

Problem: Given a general non homogeneous set of linear equations \(Ax=b\) how to test if it has no solution (inconsistent), or one unique solution, or an infinity number of solutions?

The following algorithm summarizes all the cases

Let [A|b] be the augmented matrix, where b is appended to A. 
 
Assume A is an M by N matrix. i.e. M equations and N unknowns. 
 
IF rank A < rank [A|b]  THEN  -- system is inconsistent 
   -- NO exact solution exist, but can use least square approximation 
 
   x= A/b  -- Matlab. 
   x= PseudoInverse[A].b  -- Mathematica uses SVD to computer pseduoInverse 
 
ELSE -- we must have rank A == rank[A|b]  -- system is consistent 
     IF rank(A) == N  -- we have one solution. 
        IF M==N -- one unique solution, can use crammer rule. 
           x=A/b  -- Matlab. Here we get exact solution from \ operator 
           x=LinearSolve[A,b] -- Mathematica 
        ELSE  -- infinite solutions, pick one. 
           x=A/b  -- Matlab. Here we get exact solution from \ operator 
           x=LinearSolve[A,b] -- Mathematica 
        END 
     ELSE 
        -- rank A  < N, i.e. rank deficient, infinite solutions. will pick one 
        x=A/b 
        x=LinearSolve[A,b] 
     END 
END

Let the system of equations be \begin {align*} y & =x-1\\ y & =2x+1 \end {align*}

So \[ A=\left ( {\begin {array} [c]{cc}1 & 1\\ -2 & 1 \end {array}} \right ) \] and \[ b=\left ( {\begin {array} [c]{c}1\\ -1 \end {array}} \right ) \]

Mathematica

Remove["Global`*"]; 
ContourPlot[{y==x-1,y==2 x+1} , 
            {x,-3,2},{y,-6,3}, 
     ContourStyle->{{Red,Thickness[0.01]}, 
                    {Blue,Thickness[0.01]}}, 
     GridLinesStyle->Dashed, 
     ImageSize->300]
 

a = {{-2,1},{-1,1}}; 
b = {1,-1}; 
{nRow, nCol} = Dimensions[a]; 
aRank=MatrixRank[a] 
 
   2
 

abRank=MatrixRank[Insert[a,b,-1]] 
 
   2
 

The above algorithm can now be run as follows

If[aRank<abRank, 
  Print["System is no consistent, no exact solution"]; 
  x=PseudoInverse[a].b, 
  If[aRank==nCol, 
   Print["System is consistent. Exact solution."]; 
   x=LinearSolve[a,b] 
   , 
   Print["consistent, rank deficient,infinite solutions"]; 
   x=LinearSolve[a,b] 
  ] 
]; 
 
Print["Solution is x=",x];
 

The output of the above is

System is consistent. Exact solution. 
Solution is x={-2,-3}
 

Matlab

A=[-2 1;-1 1]; 
b=[1; -1]; 
 
[nRow,nCol]=size(A); 
aRank=rank(A); 
abRank=rank([A b]); 
 
fprintf('A rank=%d, [A b] rank=%d\n',aRank,abRank); 
fprintf('Number of unknowns=%d\n',nCol); 
 
x=A\b; 
 
if aRank<abRank 
 fprintf('System not consistent. no exact solution\n'); 
else 
 if aRank==nCol 
  fprintf('System consistent. Exact solution.\n'); 
 else 
  fprintf('consistent,rank deficient,infinite solutions\n'); 
 end 
end 
 
fprintf('solution is\n'); 
x
 

Output is

A rank=2, [A b] rank=2 
Number of unknowns=2 
 
System is consistent. Exact solution. 
solution is 
x = 
    -2 
    -3