Let A be a 2 2 matrix with real entries. Let I be the 2 2 identity matrix. Denote by tr ( A ) , the sum of diagonal entries of A . Assume that A^2=1 . Statement -1: If A 1 and A -1 , then det A=-1 . Statement -2 : If A 1 and A -1 , then tr (A) 0 .

Step-by-step solution

Let A= [ array ll a & b c & d array ] so that A^2= [ array ll a^2+b c & a b+b d a c+d c & b c+d^2 array ]= [ array ll 1 & 0 0 & 1 array ] a^2+b c=1=b c+d^2 and (a+d) c=0=(a+d) b . Since A I, A 1, a=-d and hence det A= | array cc 1-b c & b c & - 1-b c array |=-1+b c-b c=-1 Statement 1 is true. But tr. A=0 and hence statement 2 is false.