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

Question:

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

(P) If $A \neq I_{2}$, then $|A|=-1$

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

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

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

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

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

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

Correct Option: , 4

Solution:

$|\mathrm{A}| \neq 0$

For $(\mathrm{P}): \mathrm{A} \neq \mathrm{I}_{2}$

So, $A=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]$ or $\left[\begin{array}{ll}1 & 1 \\ 1 & 0\end{array}\right]$ or $\left[\begin{array}{ll}0 & 1 \\ 1 & 1\end{array}\right]$ or $\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]$

or $\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right]$

$|\mathrm{A}|$ can be $-1$ or 1

So (P) is false.

For $(Q) ;|A|=1$

$A=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$ or $\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]$ or $\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right]$

$\Rightarrow \operatorname{tr}(\mathrm{A})=2$

$\Rightarrow \mathrm{Q}$ is true