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Let $A$ be a $2×2$ real matrix with entries from $\{0, 1\}$ and $|A|\neq0$. Consider the following two statements :

(P)$\quad \text{If }A \neq I_2, \text{then }|A|=-1$

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

where $I_2$ denotes $2×2$ identity matrix and $tr(A)$ denotes the sum of the diagonal entries of $A$. Then :

(1) (P) is true and (Q) are false

(2) Both (P) and (Q) are true

(3) Both (P) and (Q) are false

(4) (P) is false and (Q) is true

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Ans. (2) Both (P) and (Q) are true

Sol. Let $A = \begin{bmatrix}a&b\\c&d\end{bmatrix}\quad a, b, c, d \in \{0, 1\}$

$|A| = ad - bc \neq 0$

$\implies ad = 1,\ bc = 0 \quad\text{or}\quad ad = 0,\ bc = 1$

(P) If $A \neq I_2 \implies ad \neq 1$

$\implies ad = 0,\ bc = 1$

$\implies |A| = –1\quad$ (P) is true.

(Q) If $A = I \implies ad = 1$

$\implies ad = 1,\ bc = 0$

$\implies tr(A) = 2\quad$ (Q) is true.

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