Question

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

(P) If $A\ne I_2$then $\left|A\right|=-1$

(Q) If $\left|A\right|=1$ , then $tr\left(A\right)=2$,

where ${\text{I}}_2$ denotes $2\times 2$ identity matrix and $tr\left(A\right)$ denotes the sum of the diagonal entries of $A$. Then:

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

Answer 1

The correct option is D

(P) is false and (Q) is true

Explanation for correct option:

Finding the true statements:

$A$ be a $2\times 2$real matrix with entries from $\left\{0,1\right\}$ and $A\ne 0$

Explanation for statement (P) : If $A\ne I_2$then $\left|A\right|=-1$

Let we choose $A=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]\ne {\text{I}}_2$

The determinan of matrix $A$ is $1\left[\because 1\left(1\right)-1\left(0\right)=1\right]$.

Therefore (P) is false. since $\left|A\right|=1$ not $-1$

Explanation for statement (Q) : If $\left|A\right|=1$ , then $tr\left(A\right)=2$,

$tr\left(A\right)$ denotes the sum of the diagonal entries of $A$.

$A=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]=(1+1)=2$

$\Rightarrow$$tr\left(A\right)=2$

Therefore, (Q) is True.

Hence, option (D) is the correct answer