GATE BT 2016 • Q33 NAT • 1 Mark

Section 1: Engineering Mathematics

Linear Algebra

Subtopic: Fundamentals of Linear Algebra

Question Statement:

Match the type of chromosomal inheritance ( Column-I) with the corresponding genetic disease or trait (Column-II). Column-I Column-II P. Autosomal recessive inheritance 1. Huntington disease Q. Autosomal dominant inheritance 2. Hairy ears R. X-linked inheritance 3. Cystic fibrosis S. Y-linked inheritance 4. Hemophilia (A) P-1, Q-4, R-3, S-2 (B) P-4, Q-3, R-2, S-1 (C) P-3, Q-1, R-4, S-2 (D) P-4, Q-2, R-3, S-1

Official Answer Key:

3.0

Step-by-Step Solution & Explanation

### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2016 Q.33)** **1. Governing Concept & Definition**: - **Topic**: Engineering Mathematics & Matrix Eigenvalues - For matrix $A = \begin{pmatrix} 2 & 1 \\ 5 & -2 \end{pmatrix}$, eigenvalues satisfy the characteristic equation: $\det(A - \lambda I) = 0 \implies \lambda^2 - \operatorname{Tr}(A)\lambda + \det(A) = 0$ **2. Detailed Step-by-Step Calculation**: - **Trace and Determinant**: $\operatorname{Tr}(A) = 2 + (-2) = 0$ $\det(A) = (2)(-2) - (1)(5) = -4 - 5 = -9$ - **Characteristic Equation**: $\lambda^2 - 0\lambda - 9 = 0 \implies \lambda^2 = 9 \implies \lambda = \pm 3$ - The positive eigenvalue is $+3$. **3. Final Answer**: - **Calculated Value**: **3.0** - **Official GATE Accepted Range**: **3.0 to 3.0**

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