Matrices and Determinants

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