Question

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

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

Answer 1

Answer: (B)

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