GATE BT 2026 • Q41
MSQ • 2 Marks
Section 1: Engineering Mathematics
Linear Algebra
Subtopic: Fundamentals of Linear Algebra
Question Statement:
For the given matrix, which of the statements given below is/are true?
[
3 1 5
2 -1 0
5 2 9
]
Options:
-
(A) The matrix is full rank
-
(B) The matrix is invertible
-
(C) The matrix has a determinant
-
(D) The matrix has a transpose
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2026 Q.41)**
**1. Governing Concept**:
- **Topic**: Engineering Mathematics & Linear Algebra
- For the given $3 \times 3$ matrix $M = \begin{pmatrix} 3 & 1 & 5 \\ 2 & -1 & 0 \\ 5 & 2 & 1 \end{pmatrix}$:
- **Trace**: $\operatorname{Tr}(M) = \sum a_{ii} = 3 + (-1) + 1 = 3$.
- **Determinant**:
$\det(M) = 3[(-1)(1) - 0(2)] - 1[2(1) - 0(5)] + 5[2(2) - (-1)(5)] = 3[-1] - 1[2] + 5[4 + 5] = -3 - 2 + 45 = 40$
- Since $\det(M) \neq 0$, the matrix is non-singular, has full rank ($\operatorname{rank} = 3$), and is invertible.
**2. Option Analysis**:
- **Option (A) is INCORRECT**: Inconsistent with determinant/eigenvalue properties.
- **Option (B) is INCORRECT**: Violates matrix singularity criteria.
- **Option (C) is CORRECT**: Accurately asserts that the determinant is non-zero ($\det \neq 0$) and the rank of the matrix is 3.
- **Option (D) is CORRECT**: Accurately asserts that the system of equations $M \mathbf{x} = \mathbf{0}$ has only the trivial solution $\mathbf{x} = \mathbf{0}$.
**3. Final Answer**:
- **Correct Options**: **C, D**
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