Question

Let A be a $2\times 2$ real matrix with entries from $\{0,1\}$ and $|A|\ne 0.$ Consider the following two statements :
(P) If $A\ne I_2,$ then $|A|=-1$
(Q) If $|A|=1,$ then $tr(A)=2$
where $I_2$ denotes $2\times 2$ identity matrix and $tr(A)$ denotes the sum of the diagonal entries of $A$. Then:

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

Answer 1

Answer: (D)

$\left|A\right|\ne 0$
For $\left(P\right):A\ne I_2$
So, $A=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]or\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]or\left[\begin{array}{cc}0& 1\\ 1& 1\end{array}\right]or\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$
or $\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]$
$|A|$ Can be $-1$ or $1$
So (P) is false.
For (Q) ; $|A|=1$
$A=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]or\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]or\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]$
$\Rightarrow tr(A)=2$
$\Rightarrow Q$ is true