Let A be a `2xx2` matrix with real entries. Let I be the `2xx2` identity matrix. Denote by tr (A), the sum of diagonal entries of A. Assume that `A^2=""I` . Statement 1: If `A!=I` and `A!=""-I` , then det `A""=-1` . Statement 2: If `A!=I` and `A!=""-I` , then `t r(A)!=0` . (1) Statement 1 is false, Statement `( 2) (3)-2( 4)` is true (6) Statement 1 is true, Statement `( 7) (8)-2( 9)` (10) is true, Statement `( 11) (12)-2( 13)` is a correct explanation for Statement 1 (15) Statement 1 is true, Statement `( 16) (17)-2( 18)` (19) is true; Statement `( 20) (21)-2( 22)` is not a correct explanation for Statement 1. (24) Statement 1 is true, Statement `( 25) (26)-2( 27)` is false.

Options

A. Statement -1 is true, Statement-2 is true, Statement-2 is a
correct explanation for Statement-1
B. Statement -1 is true, Statement - 2 is true, Statement -2 is not
a correct explanation for Statement-1
C. Statement-1 is true, Statement-2 is false
D. Statement-1 is false, Statement-2 is true

Answer

C

Explanation

Let `A = [[1,0],[0,-1]]or [[-1,0],[0,1]]`
Then `A^(2) = I`
`therefore A = abs((1,0),(0,-1))= - 1 and tr (A)=0`