Problem 2. Starting with the identity matrix $I$ of shape $n\times n$ and a invertible matrix $X\in {\mathbb{C}}^{n^2\times n^2}$. Now if we say that $X$ satisfies the Yang-Baxter equation[1] :

$$(X\otimes I)(I\otimes X)(X\otimes I)=(I\otimes X)(X\otimes I)(I\otimes X)$$

then $X^{†}$ satisfies the equation (as well as $X^{-1}$). Now if take an invertable matrix $Q\in {\mathbb{C}}^{n\times n}$ and express a new matrix as:

$$X^{\mathrm{\prime }}=(Q\otimes Q)X(Q\otimes Q{)}^{-1}$$

which also satisfies the Yang-Baxter equation. Now show that given the matrix:

$$X=\frac{1+\mathit{i}}{2}\left(\begin{array}{cccc}1& 0& 0& 1\\ 0& 1& 1& 0\\ 0& -1& 1& 0\\ -1& 0& 0& 1\end{array}\right)$$

will satisfy the Yang-Baxter equation when $I$ has size $2\times 2$.

Solution 2. The first thing to do is construct the matrices $X$ and $I_2$

size of $I$ is $2\times 2$

Now create a function corresponding to the Yang-Baxter equation.

evaluate the symmetry:

The matrices given by the tensor products is: