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 $2 \times 2$ identity matrix and $\operatorname{tr}(A)$ denotes the sum of the diagonal entries of $A$. Then :
Correct Option: , 4
Let $A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$, where $a, b, c, d \in\{0,1\}$
$\Rightarrow|A|=a d-b c \neq 0$
$\Rightarrow$ either $a d=1, b c=0$ or $a d=0$ and $b c=1$
(P) If $A \neq I_{2} \Rightarrow a d \neq 1$
$\Rightarrow a d=0$ and $b c=1 \Rightarrow|A|=-1$
$\therefore \mathrm{P}$ is true.
(Q) If $|A|=1 \Rightarrow a d=1$
$\Rightarrow a d=1$ and $b c=0$
$\Rightarrow \operatorname{tr}(A)=2$
$\therefore \mathrm{Q}$ is true.
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