The system of equations  −2x+y+z=a,x−2y+z=b,x+y−2z=c  is inconsistent, if

We have  Δ= -2111-2111-2=-24-1-1-2-1+11+2=0The system of equation is inconsistent if  −2      1          −a1          −2      −b1           1            −c≠0⇒-2[2c+b]-1[-c+b]-a(3)≠0⇒-4c-2b+c-b-3a≠0⇒-3[a+b+c]≠0⇒a+b+c≠0