Question

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

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

Answer 1

Answer: (B)

$\left|A^{- 1} - k I\right|=0$
$\left|A\right|\left|A^{- 1} - k I\right|=0\left(\left|A\right| \neq 0\right)$
$\left|I - k A\right|=0$
$\left|\frac{I}{k} - A\right|=0\Rightarrow \left|A - \frac{1}{k} \cdot I\right|=0$
$\Rightarrow|A-\lambda I|=0$, 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} $