Question

The system of equations \[ x-2y+3z=6,\quad 3x+y-4z=-7,\quad 5x-3y+2z=5 \] has:

• A. No solution
• B. Unique solution
• C. Infinite number of solutions
• D. Zero solution

Hint:

If $\det(A)=0$, system may still be consistent → check rank.

Answer 1

The Correct Option is C

Concept: For a system $AX=B$: - If $\det(A)\neq 0$ → unique solution - If $\det(A)=0$ → check consistency → infinite or no solution

Step 1: Write coefficient matrix.
\[ A= \begin{bmatrix} 1&-2& 3 \\ 3& 1&-4 \\ 5&-3& 2 \end{bmatrix} \]

Step 2: Check determinant.
Expanding: \[ \det(A)= 1\begin{vmatrix} 1&-4 \\ -3& 2 \end{vmatrix} +2\begin{vmatrix} 3&-4 \\ 5& 2 \end{vmatrix} +3\begin{vmatrix} 3& 1 \\ 5&-3 \end{vmatrix} \] \[ =1(2-12)+2(6+20)+3(-9-5) \] \[ =-10+52-42=0 \] So system is dependent.

Step 3: Check consistency.
Since equations reduce to same plane relation, system is consistent and dependent. \[ \Rightarrow \text{Infinite solutions} \]