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

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 ne Iand A ne -I` then det `A=-I`
Statement-2 If `A ne I and A ne - 1`, then `tr (A) ne 0.`

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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