Let $A$ be a $2 \times 2$ real matrix with entries from $\{0, 1\}$ and $|A| \neq 0$. Consider the following two statements:

(P) If $A \neq I_2$, then $|A| = -1$

(Q) If $|A| = 1$, then $\text{tr}(A) = 2$

where $I_2$ denotes the $2 \times 2$ identity matrix and $\text{tr}(A)$ denotes the sum of the diagonal entries of $A$. Then: