Question

The value of \( a \) for which the system \[ ax + y + z = 0,\; x + ay + z = 0,\; x + y + z = 0 \] has non-zero solutions is

• A. 1,2
• B. 1,-1
• C. 1
• D. None of these

Hint:

Repeated rows/columns often give perfect square determinant.

Answer 1

The Correct Option is C

Concept: Homogeneous system has non-trivial solution when determinant = 0.

Step 1: Form determinant.
\[ \begin{vmatrix} a & 1 & 1 \\ 1 & a & 1 \\ 1 & 1 & 1 \end{vmatrix} = 0 \]

Step 2: Expand.
\[ = a(a\cdot1 - 1\cdot1) - 1(1\cdot1 - 1\cdot1) + 1(1\cdot1 - a\cdot1) \] \[ = a(a - 1) - 0 + (1 - a) = a^2 - a + 1 - a = a^2 - 2a + 1 \] \[ = (a - 1)^2 = 0 \Rightarrow a = 1 \]