Matrices and Determinants Part 1

Let A=[

a	b
c	d

]‌‌ then A2=1
⇒a2+bc=1‌‌ab+bd=0
ac+cd=0‌‌bc+d2=1
From these four relations,
a2+bc=bc+d2⇒a2=d2
and b(a+d)=0=c(a+d)
⇒a=−d
We can take a=1,b=0,c=0,d=−1 as one possible set of values, then
A=[

1	0
0	−1

]
Clearly A≠I and A≠−I and A=−1
∴‌‌ Statement 1 is true.
Also if A≠I‌‌tr(A)=0
∴‌‌ Statement 2 is false.