I see you've uploaded a set of mathematical problems. I'll address these one by one. Let’s begin:

(i) Find x and y:

We have the equation: $2\begin{bmatrix}1 & 3 \\ 0 & x \end{bmatrix} + y\begin{bmatrix}0 & 0 \\ 1 & 2\end{bmatrix} = \begin{bmatrix}5 & 6 \\ 1 & 8\end{bmatrix}.$

• Expand the left-hand side: $2\begin{bmatrix}1 & 3 \\ 0 & x\end{bmatrix} = \begin{bmatrix}2 & 6 \\ 0 & 2x\end{bmatrix}, \quad y\begin{bmatrix}0 & 0 \\ 1 & 2\end{bmatrix} = \begin{bmatrix}0 & 0 \\ y & 2y\end{bmatrix}.$

  Adding: $\begin{bmatrix}2 & 6 \\ 0 & 2x\end{bmatrix} + \begin{bmatrix}0 & 0 \\ y & 2y\end{bmatrix} = \begin{bmatrix}2 & 6 \\ y & 2x + 2y\end{bmatrix}.$

  Setting this equal to $\begin{bmatrix}5 & 6 \\ 1 & 8\end{bmatrix}$: $\begin{bmatrix}2 & 6 \\ y & 2x + 2y\end{bmatrix} = \begin{bmatrix}5 & 6 \\ 1 & 8\end{bmatrix}.$

• Solve for $x$ and $y$:

  • From $y = 1$.
  • From $2x + 2y = 8$: $2x + 2(1) = 8 \implies 2x = 6 \implies x = 3$.

Solution: $x = 3$, $y = 1$.

Would you like me to solve the rest of the problems step by step? If so, let me know.
Here are 5 related questions:

1. Can you generalize this system-solving approach for larger matrices?
2. How do elementary row transformations help find solutions to systems of equations?
3. What properties of symmetric matrices can simplify calculations?
4. How would the problem change if the matrices involved were 3x3?
5. What are some practical applications of solving systems involving matrix equations?

Tip: Always verify your results by substituting them back into the original equation!