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