If S is the set of distinct values of 'b' for which the following system of linear equations x+y+z=1, x+ay+z=1, ax+by+z=0 has no solution, then S is an empty set, a finite set containing two or more elements, an infinite set, or a singleton set?

The given set of equations can be written in matrix form as:
⎡⎢⎣1111a1ab1⎤⎥⎦[xyz]=[110]
Since this is a non-homogeneous equation, the determinant of the coefficient matrix should be 0 for no solution to exist.
∴∣∣∣∣1111a1ab1∣∣∣∣=0
⇒1(a−b)−1(1−a)+1(b−a2)=0
⇒a−b−1+a+b−a2=0
⇒2a−a2−1=0
⇒a2−2a+1=0
⇒(a−1)2=0
⇒a=1
For a=1, the equations become:
x+y+z=1
x+y+z=1
x+by+z=0
From the above three equations, we can see that if b=1, the system will be inconsistent and hence will produce no solution. For b≠1, the system will produce infinite solutions.
Hence, for no solution, S has to be a singleton set {1}.