Part CCSIR NET December 2024a-zero-on-the-border-kills-definiteness

A zero on the border kills definiteness

For a 4 × 4 positive definite real symmetric matrix A and real numbers a, b, c, d, consider the 5 × 5 matrix B whose first row is (0, a, b, c, d), whose first column is (0, a, b, c, d)ᵗ, and whose lower-right 4 × 4 block is A. Which of the following statements are necessarily true?

A.det(B) > 0 for every nonzero $(a, b, c, d) \in \mathbb{R}^{4}$ .
B.det(B) > 0 for infinitely many $(a, b, c, d) \in \mathbb{R}^{4}$ .
C.det(B) ≤ 0 for every $(a, b, c, d) \in \mathbb{R}^{4}$ .✓ Correct
D.det(B) ≤ 0 for infinitely many $(a, b, c, d) \in \mathbb{R}^{4}$ .✓ Correct

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests boundary and endpoint.

See pricing

50 are analysed free — try those first.

The trap it tests

Boundary and endpoint

The statement turns at the edge of the interval, the domain, or the parameter range.

Drill statements like this

A zero on the border kills definiteness

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