Question

Assertion :Statement- 1 : The inverse of $\left[\begin{array}{cc}\alpha & -1\\ \beta & 1\end{array}\right]$exists, where $\alpha$ and $\beta$ are the roots of the quadratic equation $x^2-2x-3=0.$ Reason: Statement- 2 : $\alpha +\beta \ne 0$

• A. Statement -1 is True, Statement -2 is True ; Statement -2 is a correct explanation for Statement -1
• 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 False
• D. Statement -1 is False, Statement -2 is True

Answer 1

The correct option is A Statement -1 is True, Statement -2 is True ; Statement -2 is a correct explanation for Statement -1

Since, $\alpha$ and $\beta$ are the roots of the equation $x^2-2x-3=0$
$\alpha +\beta =2,\alpha \beta =-3$
Given $A=\left[\begin{array}{cc}\alpha & -1\\ \beta & 1\end{array}\right]$
$\Rightarrow \left|A\right|=\alpha +\beta =2\ne 0$
Hence, $A^{-1}$ exists.