Please solve this in English please

A relation $R$ is symmetric if, for any integers $a$ and $b$, whenever $aRb$ is true, then $bRa$ must also be true.

The relation is $aRb \iff a^2 + b^2 = (a+b)^2 - 2ab$.
Let's expand the right side of the equation:

So, the relation $aRb$ simplifies to $a^2 + b^2 = a^2 + b^2$. This statement is always true for any integers $a$ and $b$.

If $aRb$ is true, it means $a^2 + b^2 = a^2 + b^2$, which is always true.
Now consider $bRa$. According to the definition, $bRa \iff b^2 + a^2 = (b+a)^2 - 2ba$.
Since addition is commutative ($b^2 + a^2 = a^2 + b^2$ and $(b+a)^2 - 2ba = (a+b)^2 - 2ab$), the expression for $bRa$ is identical to the expression for $aRb$.
Thus, $bRa \iff a^2 + b^2 = a^2 + b^2$, which is also always true.
Since $aRb$ implies $bRa$ (as both are always true), the relation $R$ is symmetric.

10. (ii) Find the inverse of the matrix $\begin{pmatrix} 2 & -3 & 2 \\ 1 & 4 & -1 \\ 5 & 2 & 1 \end{pmatrix}$. Hence solve the system of equations $\begin{cases} 2x - 3y + 2z = 2 \\ x + 4y - z = -4 \\ 5x + 2y + z = -2 \end{cases}$

Let the given matrix be $A = \begin{pmatrix} 2 & -3 & 2 \\ 1 & 4 & -1 \\ 5 & 2 & 1 \end{pmatrix}$.
The system of equations can be written as $AX = B$, where $X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$ and $B = \begin{pmatrix} 2 \\ -4 \\ -2 \end{pmatrix}$.
To solve for $X$, we need to find $A^{-1}$ and then calculate $X = A^{-1}B$.

The cofactor matrix is:

Calculate each component:

The solution to the system of equations is $\boxed{x=1, y=-2, z=-3}$.