Let A, other than I or - I, be a $$2 \times 2$$ 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

Let $$A = \begin{bmatrix} \alpha & \beta \\ \gamma & \delta \end{bmatrix}$$

$$A^2 = \begin{bmatrix} \alpha^2 + \beta\gamma & \beta(\alpha + \delta) \\ \gamma(\alpha + \delta) & \delta^2 + \beta\gamma \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$

$$\alpha + \delta = 0$$ and $$\alpha^2 + \beta\gamma = 1$$

$$\text{Tr}(A) = 0$$

$$\text{det}A = \alpha\delta - \beta\gamma = -\alpha^2 - \beta\gamma = -(\alpha^2 + \beta\gamma) = -1$$

Thus, statement-1 is true but statement-2 is false.