Question

Let $A$ be a $2\times 2$ matrix with non-zero entries and let $A^2$ = $I$, where $I$ is $2\times 2$ identity matrix. Define $Tr(A)$ = Sum of diagonal elements of $A$ and $|A|$ = determinant of matrix $A$.
Statement-1. $Tr(A)=0$
Statement-2. $|A|=1$

• A. Statement-1 is true, statement-2 is true ; statement-2 is not correct explanation for statement-1.
• B. Statement-1 is true, statement-2 is false
• C. Statement-1 is false, statement-2 is true.
• D. ) Statement-1 is true, statement-2 is true ; statement-2 is correct explanation for statement-1.

Answer 1

Answer: (B)

Let A = $\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ $\therefore$ adj A =$\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$
$\Rightarrow$ adj (adj A) = $\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=A$ $\therefore$ Statement-1 is true.
Also | adj A | = ad - bc = 1 | A |
$\therefore$ Statement-2 is true.
Since Statement-2 does not give Statement-1