Question

The possible values of scalar $k$ such that the matrix $A^{-1}-kI$ is singular, where $A=\left[\begin{array}{ccc}1& 0& 2\\ 0& 2& 1\\ 1& 0& 0\end{array}\right],$ are

• A. $\frac{-1}{2},2$
• B. $-1,\frac{1}{2}$
• C. $\frac{1}{2},\frac{-1}{2}$
• D. $-1,1$

Answer 1

Answer: (B)

$\left|A^{-1}-kI\right|=0$
$\left|A\right|\left|A^{-1}-kI\right|=0\left(\left|A\right|\ne 0\right)$
$\left|I-kA\right|=0$
$\left|\frac{I}{k}-A\right|=0\Rightarrow \left|A-\frac{1}{k}\cdot I\right|=0$
$\Rightarrow |A-\lambda I|=0$, where $\lambda =\frac{1}{k}$
$\Rightarrow \left|\begin{array}{ccc}1-\lambda & 0& 2\\ 0& 2-\lambda & 1\\ 1& 0& -\lambda \end{array}\right|=0$
$\begin{array}{l}\Rightarrow (1-\lambda )(-\lambda )(2-\lambda )+2(0-(2-\lambda ))=0\\ \Rightarrow -{\lambda }^3+3{\lambda }^2-2\lambda -4+2\lambda =0\\ \Rightarrow {\lambda }^3-3{\lambda }^2+4=0\Rightarrow \lambda =2,2,-1\Rightarrow k=-1,\frac{1}{2}\end{array}$