Let A be a `2 xx 2` matrix with non-zero entries and let A^2=I, where i is a `2 xx 2` identity matrix, Tr(A) i= sum of diagonal elements of A and `|A|` = determinant of matrix A. Statement 1:Tr(A)=0 Statement 2:`|A|`=1 . then (A) Statement 1 is false, statement 2 is true. (B) Statement 1 is true, statement 2 is true, statement 2 is a correct explanation for statement 1. (C) Statement 1 is true, statement 2 is true, statement 2 is a correct explanation for statement 1. (D) Statement 1 is true, statement 2 is false.

Options

A. Statement 1 is false, statement 2 is true.
B. Statement 1 is true, statement 2 is true, statement 2 is a correct explanation for statement 1.
C. Statement 1 is true, statement 2 is true, statement 2 is a correct explanation for statement 1.
D. Statement 1 is true, statement 2 is false.

Answer

D

Explanation

Let `A=[(a,b),(c,d)], a, b, c, d ne 0`
`A^(2)=[(a,b),(c,d)]xx[(a,b),(c,d)]=[(a^(2)+bc,ab+bd),(ac+cd,bc+d^(2))]`
Given `A^(2)=I`
`implies a^(2)+bc=1, bc+d^(2)=1`
`ab+bd=ac+cd=0`
`c ne 0` and `b ne 0implies a+d=0`
Trace `A=a+d=0`
`:. |A|=ad-bc=-a^(2)-bc=-1`