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. Tr (A) = 0 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)

Statement-1 is true; Statement-2 is true; Statement-2 is not a correct explanation for Statement-1

Answer 2

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