The values of  'x'  for which the matrix  is x+abcax+bcabx+c  non-singular are

x+abcax+bcabx+cC1→C1+C2+C3 (x+a+b+c)1bc1x+bc1bx+cR2→R2−R1R3→R3−R1(x+a+b+c)1bc0x000x=x2(x+a+b+c)Given matrix is  non singular⇒x2(x+a+b+c)≠0⇒  x≠0,  x≠−a−b−c