Question Statement:
The following system of linear equations
7x - 3y + z = 0
3x - y + z = 0
x - y - z = 0
has:
Options:
-
(A) infinitely many solutions
-
(B) a unique solution
-
(C) no solution
-
(D) three solutions
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE AE 2024 Q.11)**
**1. Governing Concept & Syllabus Domain**:
This question pertains to **Linear Algebra & Differential Equations** under *Engineering Mathematics* (1 Mark, MCQ).
**2. Linear System Analysis**:
- System of homogeneous linear equations:
$7x - 3y + z = 0, \quad 3x - y + z = 0, \quad x - y - z = 0$
- Evaluate determinant of coefficient matrix $A$:
$\det(A) = \det\begin{bmatrix} 7 & -3 & 1 \\ 3 & -1 & 1 \\ 1 & -1 & -1 \end{bmatrix} = 7(1 + 1) - (-3)(-3 - 1) + 1(-3 + 1) = 14 - 12 - 2 = 0$
- Since $\det(A) = 0$ for a homogeneous system $AX = 0$, there exist non-trivial solutions, meaning **infinitely many solutions**.
**3. Final Answer**:
Correct Option is **(A)** (infinitely many solutions).
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