Question

Let $P$ be a $2\times 2$ matrix such that $\left[\begin{array}{cc}1& 0\end{array}\right]P=-\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& 1\end{array}\right]$ and $\left[\begin{array}{cc}0& 1\end{array}\right]P=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}-1& 1\end{array}\right]$
If $0$ and $I$ denote the zero and identity matrices of order $2$, respectively, then which of the following options is correct ?

• A. $P^8-P^6+P^4+P^2=0$
• B. $P^8+P^6-P^4+P^2=I$
• C. $P^8+P^6+P^4-P^2=2I$
• D. $P^8-P^6-P^4-P^2=0$

Answer 1

The correct option is C $P^8+P^6+P^4-P^2=2I$

$\left[\begin{array}{cc}1& 0\end{array}\right]\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=-\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& 1\end{array}\right]$
$\Rightarrow \left[\begin{array}{cc}a& b\end{array}\right]=-\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& 1\end{array}\right]$

$\left[\begin{array}{cc}0& 1\end{array}\right]\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}-1& 1\end{array}\right]$

$\Rightarrow \left[\begin{array}{cc}c& d\end{array}\right]=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}-1& 1\end{array}\right]$
$\Rightarrow \left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\left[\begin{array}{cc}-\frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\\ -\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\end{array}\right]\Rightarrow P=-\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& 1\\ 1& -1\end{array}\right]\Rightarrow P^2=\frac{1}{2}\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]P^2=I\Rightarrow P^4,P^6,P^8=I\Rightarrow P^8+P^6+P^4-P^2=2I$