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Re: Strange empty set of solutions


José Carlos Santos wrote:
> Hi all,
> 
> I have a certain 3 x 3 numerical matrix M. If I type
> 
> Solve[M.{a,b,c}=={0,0,0},{a,b,c}]
> 
> I get
> 
> {{a -> 0. + 0.0410026 c, b -> 0. + 1.35294 c}}
> 
> However, if I type
> 
> Solve[{M.{a,b,c}=={0,0,0},a^2+b^2+c^2==1},{a,b,c}]
> 
> I get the empty set. Why is that? 

According to the online help, "Solve gives {} if there are no possible 
solutions to the equations. "

> I should get two solutions!

Not knowing your matrix M, I cannot tell whether your expectation are 
legitimate; but here is a simple system that returns several solutions:

In[1]:=
m = {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}};

In[2]:=
sol = Solve[m . {a, b, c} == {0, 0, 0}, {a, b, c}]

 From In[2]:=
Solve::svars: Equations may not give solutions for all "solve" variables.

Out[2]=
{{a -> c, b -> -2*c}}

In[3]:=
a^2 + b^2 + c^2 == 1 /. sol

Out[3]=
{6*c^2 == 1}

In[4]:=
sol = Solve[m . {a, b, c} == {0, 0, 0} &&
     a^2 + b^2 + c^2 == 1, {a, b, c}]

Out[4]=
{{a -> -(1/Sqrt[6]), b -> Sqrt[2/3],
    c -> -(1/Sqrt[6])}, {a -> 1/Sqrt[6],
    b -> -Sqrt[2/3], c -> 1/Sqrt[6]}}

Regards,
Jean-Marc


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