Question

Which of the following matrices do not have eigenvalues as 1 and -1?

• A. $\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$
• B. $\left[\begin{array}{cc}0& -i\\ i& 0\end{array}\right]$ (where $i=\sqrt{-1}$)
• C. $\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$
• D. $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

Answer 1

The correct option is D $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

For option A,
$A=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$
The characteristic equation of A is given by
$|A-\lambda I|=0$
$\Rightarrow \left|\begin{array}{cc}-\lambda & 1\\ 1& -\lambda \end{array}\right|=0$
$\Rightarrow (\lambda {)}^2-1=0$
$\Rightarrow \lambda =-1,1$
For option B,
$A=\left[\begin{array}{cc}0& -i\\ i& 0\end{array}\right]$
The characteristic equation of A is given by
$|A-\lambda I|=0$
$\Rightarrow \left|\begin{array}{cc}-\lambda & -i\\ i& -\lambda \end{array}\right|=0$
$\Rightarrow (\lambda {)}^2-1=0$
$\Rightarrow \lambda =-1,1$
For option C,
$A=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$
The characteristic equation of A is given by
$|A-\lambda I|=0$
$\Rightarrow \left|\begin{array}{cc}1-\lambda & 0\\ 0& -1-\lambda \end{array}\right|=0$
$\Rightarrow (\lambda {)}^2-1=0$
$\Rightarrow \lambda =-1,1$
For option D,
$A=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
The characteristic equation of A is given by
$|A-\lambda I|=0$
$\Rightarrow \left|\begin{array}{cc}1-\lambda & 0\\ 0& 1-\lambda \end{array}\right|=0$
$\Rightarrow (1-\lambda {)}^2=0$
$\Rightarrow \lambda =1$
Hence, the eigen value of option D matrix is 1 only.