# Let A be a 2x2 real matrix with entries from

Question:

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 :

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

2. (2) Both (P) and (Q) are false

3. (3) (P) is true and (Q) is false

4. (4) Both (P) and (Q) are true

Correct Option: , 4

Solution:

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.