GATE AE 2020 • Q19 MCQ • 1 Mark Engineering Mathematics

Engineering Mathematics

Linear Algebra

Subtopic: Eigenvalues & Eigenvectors

Question Statement:

Given $A = \begin{bmatrix} \sin\theta & \tan\theta \\ 0 & \cos\theta \end{bmatrix}$, the sum of squares of the eigenvalues of $A$ is:

Options:

  • (A) $\sin^2\theta + \cos^2\theta = 1$
  • (B) $\tan^2\theta$
  • (C) $\sin 2\theta$
  • (D) 0

Official Answer Key:

A

Step-by-Step Solution & Explanation

### **Detailed Step-by-Step Solution (GATE AE 2020 Q.19)** **1. Intuitive Physical Setup & Given Parameters**: - Matrix $A = \begin{bmatrix} \sin\theta & \tan\theta \\ 0 & \cos\theta \end{bmatrix}$ is an upper triangular matrix ($a_{21} = 0$). **2. Step-by-Step Mathematical Derivation**: 1. **Eigenvalues of Upper Triangular Matrix**: The eigenvalues of any triangular matrix are simply the diagonal entries: \lambda_1 = \sin\theta, \quad \lambda_2 = \cos\theta 2. **Sum of Squares of Eigenvalues**: S = \lambda_1^2 + \lambda_2^2 = \sin^2\theta + \cos^2\theta 3. By the fundamental Pythagorean trigonometric identity: \sin^2\theta + \cos^2\theta = 1 Therefore, the sum of squares of the eigenvalues is identically **1** (independent of $\theta$). **3. Comprehensive Options & Distractor Analysis**: - **(A)** $\sin^2\theta + \cos^2\theta = 1$: ✅ **CORRECT**. Directly derived from diagonal eigenvalues and Pythagorean identity. - **(B)** $\tan^2\theta$: ❌ Incorrect. - **(C)** $\sin 2\theta$: ❌ Incorrect. That would relate to $2\lambda_1\lambda_2$. - **(D)** 0: ❌ Incorrect. **4. Official Master Key & Exam Takeaway**: - **Official Answer**: **A** - **Exam Rule**: For triangular matrices, eigenvalues are diagonal elements; $\lambda_1^2 + \lambda_2^2 = 1$.
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