Question

The possible values of scalar $\text{k}$ such that the matrix $\text{A}^{- 1} - kI$ is singular where $A=\begin{bmatrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 1 & 0 & 0 \end{bmatrix},$ 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)

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