Matrices and Determinants: JEE Main 2010 Mathematics question

Let $A$ be a $\,2 \times 2$ matrix with non-zero entries and let ${A^2} = I,$ where $I$ is $2 \times 2$ identity matrix. Define $Tr$$(A)=$ sum of diagonal elements of $A$ and $\left| A \right| =$ determinant of matrix $A$. Statement- 1: $Tr$$(A)=0$. Statement- 2: $\left| A \right| = 1$ .