Question:

<p><strong>Step 1: Write the augmented matrix</strong></p><p>The system is:</p><p>$x + y + z = 1$</p><p>$x + ay + z = 1$</p><p>$ax + by + z = 0$</p><p>Augmented matrix: $[A|B] = $</p><p>$$\begin{bmatrix}1 & 1 & 1 & | & 1\\1 & a & 1 & | & 1\\a & b & 1 & | & 0\end{bmatrix}$$</p><p><strong>Step 2: Find the coefficient matrix determinant</strong></p><p>$$\det(A) = 1(a - b) - 1(1 - a) + 1(b - a^2)$$</p><p>$$= a - b - 1 + a + b - a^2$$</p><p>$$= 2a - a^2 - 1$$</p><p>$$= -(a^2 - 2a + 1)$$</p><p>$$= -(a - 1)^2$$</p><p><strong>Step 3: Determine when the system has no solution</strong></p><p>For no solution, $\det(A) = 0$ AND the system is inconsistent.</p><p>$\det(A) = -(a - 1)^2 = 0$ implies $a = 1$</p><p><strong>Step 4: Check consistency when $a = 1$</strong></p><p>When $a = 1$, the system becomes:</p><p>$x + y + z = 1$</p><p>$x + y + z = 1$</p><p>$x + by + z = 0$</p><p>The first two equations are identical. For no solution, the third equation must be inconsistent with them.</p><p>From equations 1 and 2: $x + y + z = 1$</p><p>From equation 3: $x + by + z = 0$</p><p>Subtracting: $(1 - b)y = 1$</p><p><strong>Step 5: Find the value of $b$ for inconsistency</strong></p><p>If $1 - b \neq 0$ (i.e., $b \neq 1$), then $y = \frac{1}{1-b}$, giving a unique solution.</p><p>If $1 - b = 0$ (i.e., $b = 1$), then we get $0 = 1$, which is a contradiction.</p><p>Therefore, no solution occurs when $a = 1$ and $b = 1$.</p><p><strong>Step 6: Identify the set $S$</strong></p><p>The only value of $b$ for which the system has no solution is $b = 1$.</p><p>Therefore, $S = \{1\}$, which is a singleton set.</p><p><strong>∴ Answer: (1)</strong></p>