Let A , other than I or - I , be a 2 2 real matrix such that A ^2= I , I being the unit matrix. Let Tr ( A ) be the sum of diagonal elements of A. Statement-1: Tr ( A )=0 Statement-2: det ( A )=-1

Step-by-step solution

[ array ll a & b c & d array ] [ array ll a & b c & d array ]= [ array ll 1 & 0 0 & 1 array ] aligned & [ array ll a^2+b c & a b+b d a c+c d & b c+d^2 array ]= [ array ll 1 & 0 0 & 1 array ] & b(a+d)=0, b=0 or a=-d & c(a+d)=0, c=0 or a=-d & a^2+b c=1, b c+d^2=1 aligned ' a ' and ' d ' are diagonal elements a+d=0 statement- 1 is correct. Now, det (A)=a d-b c Now, from (3) a^2+b c=1 and d^2+b c=1 So, a^2-d^2=0 Adding a^2+d^2+2 b c=2 =(a+d)^2-2 a d+2 b c=2 or 0-2(a d-b c)=2 So, a d-b c=1 det (A)=-1 So, statement -2 is