Question 79 MCQ · 2.0 marks
A is positive definite.
A is positive semi-definite.
A is negative definite.
A is negative semi-definite.
**Java**
**Section Id :** 64065373860
**Section Number :** 6
**Section type :** Online
**Mandatory or Optional :** Mandatory
**Number of Questions :** 16
**Number of Questions to be attempted :** 16
**Section Marks :** 100
**Display Number Panel :** Yes
**Section Negative Marks :** 0
**Group All Questions :** No
**Enable Mark as Answered Mark for Review and**
No
**Clear Response :**
**Section Maximum Duration :** 0
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Published solution
**Answer: B**
**Topic:** Linear Algebra — *Positive Semi-Definiteness of a Singular 2x2 Symmetric Matrix*
### Step-by-Step Solution
**Step 1:** Examine the symmetric matrix $A = \begin{pmatrix} 1 & 4 \\ 4 & 16 \end{pmatrix}$ by computing its trace and determinant (or eigenvalues):
- $\text{tr}(A) = 1 + 16 = 17 > 0$
- $\det(A) = (1)(16) - (4)(4) = 16 - 16 = 0$
**Step 2:** Since $\lambda_1 + \lambda_2 = 17$ and $\lambda_1 \lambda_2 = 0$, the eigenvalues of $A$ are $\lambda_1 = 17 > 0$ and $\lambda_2 = 0$.
**Step 3:** Because all eigenvalues are **non-negative** ($\lambda_i \ge 0$) with at least one zero eigenvalue, $A$ is **positive semi-definite** (**B**).
**Key Takeaway / Tip:** Equivalently, $x^T A x = x_1^2 + 8x_1 x_2 + 16x_2^2 = (x_1 + 4x_2)^2 \ge 0$ for all $x \in \mathbb{R}^2$, which equals $0$ along the line $x_1 = -4x_2$, proving $A$ is positive semi-definite.
A: **Incorrect:** Option A (A is positive definite.) does not satisfy the conditions derived in the step-by-step analysis for Positive Semi-Definiteness of a Singular 2x2 Symmetric Matrix (correct choice: B).
B: **Correct:** Option B (A is positive semi-definite.) matches our derived result under Positive Semi-Definiteness of a Singular 2x2 Symmetric Matrix. Equivalently, $x^T A x = x_1^2 + 8x_1 x_2 + 16x_2^2 = (x_1 + 4x_2)^2 \ge 0$ for all $x \in \mathbb{R}^2$, which equals $0$ along the line $x_1 = -4x_2$, proving $A$ is positive semi-definite.
C: **Incorrect:** Option C (A is negative definite.) does not satisfy the conditions derived in the step-by-step analysis for Positive Semi-Definiteness of a Singular 2x2 Symmetric Matrix (correct choice: B).
D: **Incorrect:** Option D (A is negative semi-definite. **Java** **Section Id :** 64065373860 **Section Number :** 6 **Section type :** Online **Mandatory or Optional :** Mandatory **Number of Quest...) does not satisfy the conditions derived in the step-by-step analysis for Positive Semi-Definiteness of a Singular 2x2 Symmetric Matrix (correct choice: B).
