2x + 3y + z = 0
x + y = 0
y + z = 0
The given system of equations has
Options:
(A) a unique solution.
(B) infinitely many solutions.
(C) no solution.
(D) a finite number of solutions.
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2025 Q.12)**
**1. Intuitive Physical Setup & Given Parameters**:
We are given the system of homogeneous linear equations:
$2x + 3y + z = 0$
$x + y + 0z = 0$
$0x + y + z = 0$
and asked to classify its solution set.
**2. Step-by-Step Mathematical Derivation**:
1. **Matrix Representation**:
We write the system as $A\mathbf{x} = \mathbf{0}$, where the coefficient matrix is:
$A = \begin{bmatrix} 2 & 3 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \end{bmatrix}$
2. **Determinant Computation**:
$\det(A) = 2(1 \cdot 1 - 0 \cdot 1) - 3(1 \cdot 1 - 0 \cdot 0) + 1(1 \cdot 1 - 1 \cdot 0)$
$\det(A) = 2(1) - 3(1) + 1(1) = 2 - 3 + 1 = 0$
3. **Rank & Solution Dimension**:
Since $\det(A) = 0$, the rank of $A$ is $\text{rank}(A) < 3$. Inspecting the $2 \times 2$ submatrix $\begin{vmatrix} 1 & 1 \\ 0 & 1 \end{vmatrix} = 1 \ne 0$, so $\text{rank}(A) = 2$.
By the Rank-Nullity theorem, nullity $= n - \text{rank}(A) = 3 - 2 = 1 > 0$.
A homogeneous linear system with nullity $\ge 1$ possesses non-trivial solutions, and thus has **infinitely many solutions** (forming a 1-D subspace parameterized by $t$).
**3. Comprehensive Options & Distractor Analysis**:
- **(A) a unique solution**: ❌ **INCORRECT**. A unique trivial solution requires $\det(A) \ne 0$.
- **(B) infinitely many solutions**: ✅ **CORRECT**. Since $\det(A) = 0$ for a homogeneous system, there exists a one-parameter family of non-trivial solutions.
- **(C) no solution**: ❌ **INCORRECT**. Homogeneous systems are always consistent because $\mathbf{x} = \mathbf{0}$ is always a solution.
- **(D) a finite number of solutions**: ❌ **INCORRECT**. Linear systems in $\mathbb{R}$ cannot have a finite number of multiple solutions; they have either 0, 1, or $\infty$.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **B**
- **Exam Takeaway**: A homogeneous system $Ax = 0$ is ALWAYS consistent. If $\det(A) \ne 0$, it has only the unique trivial solution $(0,0,0)$. If $\det(A) = 0$, it has infinitely many solutions.
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