### **Detailed Step-by-Step Solution (GATE AE 2024 Q.11)**
**1. Intuitive Physical Setup & Given Parameters**:
We are given the homogeneous system of three linear equations in three unknowns $(x, y, z)$:
$7x - 3y + z = 0$
$3x - y + z = 0$
$x - y - z = 0$
and asked to determine the nature of its solution set.
**2. Step-by-Step Mathematical Derivation**:
1. **Construct the Coefficient Matrix**:
$A = \begin{bmatrix} 7 & -3 & 1 \\ 3 & -1 & 1 \\ 1 & -1 & -1 \end{bmatrix}$
2. **Compute the Determinant**:
$\det(A) = 7((-1)(-1) - (1)(-1)) - (-3)((3)(-1) - (1)(1)) + 1((3)(-1) - (-1)(1))$
$\det(A) = 7(1 + 1) + 3(-3 - 1) + 1(-3 + 1) = 7(2) + 3(-4) + 1(-2) = 14 - 12 - 2 = 0$
3. **Rank and Solution Space Dimension**:
Because $\det(A) = 0$, the $3 \times 3$ matrix is singular (rank
lt; 3$).
Checking the upper $2 \times 2$ minor: $\begin{vmatrix} 7 & -3 \\ 3 & -1 \end{vmatrix} = -7 - (-9) = 2 \ne 0$, so $\text{rank}(A) = 2$.
By the Rank-Nullity Theorem: $\text{Nullity} = 3 - \text{rank}(A) = 3 - 2 = 1$.
A homogeneous linear system with nullity $\ge 1$ always possesses non-trivial solutions, yielding an infinite one-parameter line of solutions through the origin.
**3. Comprehensive Options & Distractor Analysis**:
- **(A) infinitely many solutions**: ✅ **CORRECT**. Determinant is zero, so the homogeneous system has non-trivial solutions forming an infinite family.
- **(B) a unique solution**: ❌ **INCORRECT**. Requires $\det(A) \ne 0$, which would yield only $(0,0,0)$.
- **(C) no solution**: ❌ **INCORRECT**. Homogeneous systems are always consistent since $(0,0,0)$ is always a trivial solution.
- **(D) exactly two solutions**: ❌ **INCORRECT**. Linear systems over $\mathbb{R}$ cannot have a finite number of multiple solutions.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **A**
- **Exam Takeaway**: For any homogeneous system $Ax = 0$: If $\det(A) \ne 0$, unique trivial solution. If $\det(A) = 0$, infinitely many solutions.