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 $tr( A )=2$
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 = \begin{bmatrix}0&1\\ 1&0\end{bmatrix} or \begin{bmatrix}1&1\\ 1&0\end{bmatrix} or \begin{bmatrix}0&1\\ 1&1\end{bmatrix} or \begin{bmatrix}1&1\\ 0&1\end{bmatrix} $
or $\begin{bmatrix}1&0\\ 1&1\end{bmatrix}$
$|A|$ Can be $-1$ or $1$
So (P) is false.
For (Q) ; $|A| = 1$
$A = \begin{bmatrix}1&0\\ 0&1\end{bmatrix} or \begin{bmatrix}1&1\\ 0&1\end{bmatrix} or \begin{bmatrix}1&0\\ 1&1\end{bmatrix} $
$\Rightarrow tr(A) = 2$
$\Rightarrow Q$ is true