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

• A
  $(P)$ is true and $(Q)$ is false
• B
  Both $(P)$ and $(Q)$ are false
• C
  Both $(P)$ and $(Q)$ are true
• D
  $(P)$ is false and $(Q)$ is true