Question

Let A, other than I or - I, be a 2 $\times$ 2 real matrix such that $A^2=I$, I being the unit matrix. Let Tr (A) be the sum of diagonal elements of A.
Statement-1: Tr (A) = 0
Statement-2: det (A) = - 1

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

Answer 1

Answer: (B)

$\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$\left[\begin{array}{cc}{a}^2+bc& ab+bd\\ ac+cd& bc+d^2\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$b(a+d)=0,b=0$ or $a=-d$ … (1)
$c(a+d)=0,c=0$ or $a=-d$ … (2)
$a^2+bc=1,bc+d^2=1$ … (3)
‘a’ and ‘d’ are diagonal elements a + d = 0 statement-1 is correct.
Now, det(A) = ad - bc
Now, from (3) $a^2+bc=1$ and $d^2+bc=1$
So, $a^2-d^2=0$
Adding $a^2+d^2+2bc=2$
$\Rightarrow (a+d{)}^2-2ad+2bc=2$
or $0-2(ad-bc)=2$
So, ad - bc = 1 $\Rightarrow$ det(A) = -1
So, statement - 2 is also true.
But statement - 2 is not the correct explanation of statement-1.