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

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=[(a,b),(c,d)]:}a,b,c,dne 0`. Then,
`{:A=[(a,b),(c,d)][(a,b),(c,d)]=[(a^2+bc,ab+bd),(ac+dc,bc+d^2)]:}`
`:. A^2=I`
`rArr a^2+bc=1,bc+d^2=1,ac+dc=0 andab+bd=0`
`rArr (a^2+bc)-(d^2+bc)=0,c(a+d)=0and (a+d)b=0`
`rArr a^2-b^2=0 and a+d=0 [:' b ne 0, c ne 0]`
`rarr a+d=0rArr Tr(A)=0`
So, statement -1 true.
`absA={:[(a,b),(c,d)]:}=ad-bc=-a^2-bc [:'d=-a]`
`rArr absA=-1 [:' a^2+bc=1]`
So, statement -2 is not true.