ask a question to expertQuestion $3[4$ points$]$: Pauli operatorQuestion $3[4$ points$]$: Pauli operators

Recall the definition of the I,$x,Z,$ and $Y$ gates:

$I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],Z=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right],x=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right],Y=\left[\begin{array}{cc}0& -i\\ i& 0\end{array}\right]$

These are known as the Pauli matrices or gates.

Compute the matrices xoxZ and Zoxx

Show that the non$-$identity Pauli matrices anti$-$commute: that is$,UV=-VU$ for every pair

of $x,Y,$ and $Z$ matrices where $UV$

Show that the Pauli matrices I,$x,Z,Y$ are linearly independent

Show that the Pauli matrices form a basis for the space of $22$ complex$-$valued matrices.

Recall the definition of the I,$x,Z,$ and $Y$ gates:

$I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],Z=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right],x=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right],Y=\left[\begin{array}{cc}0& -i\\ i& 0\end{array}\right]$

These are known as the Pauli matrices or gates.