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)

$\begin{bmatrix}a&b\\ c &d\end{bmatrix} \begin{bmatrix}a&b\\ c&d\end{bmatrix} = \begin{bmatrix}1&0\\ 0 &1\end{bmatrix}$
$ \begin{bmatrix}a^{2} + bc&ab+bd\\ ac+cd &bc+d^{2}\end{bmatrix} = \begin{bmatrix}1&0\\ 0 &1\end{bmatrix} $
$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.