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Let A be a 2xx2 matrix with real ent...

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.

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.

Text Solution

Verified by Experts

The correct Answer is:
C
Let `{:A=[(a,b),(c,d)]:}`. Then
`A ne I and A ne -I rArr a=d ne 1 or, a=dne-1`.
It is given that `A^2=I`
`:.{:[(a,b),(c,d)][(a,b),(c,d)]=[(1,0),(0,1)]:}`
`rArr{:[(a^2+bc,ab+bd),(av+bc,bc+d^2)]=[(1,0),(0,1)]:}`
`rArr a^2+bc =1, ab+bd=0, ac+dc=0 and bc+d^2=1`
`rArr a=-d and a=pmsqrt(1-bc)`
`:. absA=ad -bc=-a^2-bc=-1+bc -bc=-1`
and, `tr (A) =a+d=0`
Hence,`tr(A) =a+d=0`
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