Question

If $A=\left[\begin{array}{ccc}2& \lambda & -3\\ 0& 2& 5\\ 1& 1& 3\end{array}\right]$, the then $A^{-1}$ exists, if

• A. $\lambda =2$
• B. $\lambda =-2$
• C. $\lambda \ne -2$
• D. None of these

Answer 1

Answer: (D)

The given matrix is $A=\left[\begin{array}{ccc}2& \lambda & -3\\ 0& 2& 5\\ 1& 1& 3\end{array}\right]$.
Now, $A^{-1}$ exists, if $|A|\ne 0$
$\Rightarrow \left|\begin{array}{ccc}2& \lambda & -3\\ 0& 2& 5\\ 1& 1& 3\end{array}\right|\ne 0$
$\Rightarrow 2\left|\begin{array}{cc}2& 5\\ 1& 3\end{array}\right|-\lambda \left|\begin{array}{cc}0& 5\\ 1& 3\end{array}\right|-3\left|\begin{array}{cc}0& 2\\ 1& 1\end{array}\right|\ne 0$
$\Rightarrow 2(6-5)-\lambda (0-5)-3(0-2)\ne 0$
$\Rightarrow 2+5\lambda +6\ne 0$
$\Rightarrow 5\lambda \ne -8$
$\Rightarrow \lambda \ne -\frac{8}{5}$