JEE Main 2013 Mathematics question

Let A, other than I or −I, be a 2 × 2 real matrix such that , I being the unit matrix. Let Tr(A) be the sum of diagonal elements of A. Statement-1: Tr(A) = 0 Statement-2: det(A) = −1

(1)Statement-1 is true; Statement- 2 is false.
(2)Statement-1 is true; Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.Correct
(3)Statement-1 is true; Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(4)Statement-1 is false; Statement- 2 is true.

Answer: (2)

Solution

means every eigenvalue is ±1; since A ≠ ±I, the eigenvalues are 1 and −1. Hence Tr(A) = 0 and det(A) = −1 — both true. (Directly: with $A =

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Question ID VK-104151