Question

The system of linear equations \[ x+y+z=0,\; 2x+y-z=0,\; 3x+2y=0 \] has

• A. no solution
• B. a unique solution
• C. infinitely many solutions
• D. None of these

Hint:

Determinant = 0 $\Rightarrow$ either no solution or infinite solutions—check consistency.

Answer 1

The Correct Option is C

Concept: If determinant = 0 and system is consistent → infinitely many solutions. 

Step 1: Form determinant.
\[ \begin{vmatrix} 1 & 1 & 1 \\ 2 & 1 & -1 \\ 3 & 2 & 0 \end{vmatrix} \] 

Step 2: Evaluate.
\[ = 1(1\cdot0 - (-1)\cdot2) - 1(2\cdot0 - (-1)\cdot3) + 1(2\cdot2 - 1\cdot3) \] \[ = 1(2) - 1(3) + 1(4 - 3) = 2 - 3 + 1 = 0 \] 

Step 3: Conclusion.
\[ \text{Determinant } = 0 \Rightarrow \text{not unique} \] Check consistency → equations are dependent \( \Rightarrow \) infinitely many solutions.