Are there any solutions?

Question

Consider the following system

 

X₁+X₂+X₃=-1

-X₁          =-1

X₁+X₂-X₃ =-1

 

Without solving the system, can you tell if there are any solutions? If so, how many?

 

Answer 1

You can evaluate the system determinant; if it's not zero, then there is one solution (an ordered triple).

 

I believe that the determinant equals -2.  Check it!