\text{If}\mathrm{ A}=[\left[\begin{array}{cc}x& -y\\ y& x\end{array}\right]\text{]and}A+A^{\prime }=2I,\text{then}x\text{is equal to}

The correct answer is 1.

Let us solve the problem step-by-step.

We are given the matrix $A$ as:
$$A=\left[\begin{array}{cc}x& -y\\ y& x\end{array}\right]$$

The transpose of matrix $A$, denoted by $A^{\prime }$ (or $A^T$), is obtained by swapping its rows and columns:
$$A^{\prime }=\left[\begin{array}{cc}x& y\\ -y& x\end{array}\right]$$

We are also given the equation:
$$A+A^{\prime }=2I$$
where $I$ is the $2\times 2$ identity matrix:
$$I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$

Substituting the matrices into the equation:
$$\left[\begin{array}{cc}x& -y\\ y& x\end{array}\right]+\left[\begin{array}{cc}x& y\\ -y& x\end{array}\right]=2\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$

Adding the matrices on the left-hand side and multiplying the matrix on the right-hand side by 2, we get:
$$\left[\begin{array}{cc}x+x& -y+y\\ y-y& x+x\end{array}\right]=\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]$$

Simplifying the elements:
$$\left[\begin{array}{cc}2x& 0\\ 0& 2x\end{array}\right]=\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]$$

By comparing the corresponding elements on both sides of the matrix equation, we get:
$$2x=2$$

Dividing by 2 on both sides:
$$x=1$$

Thus, the value of $x$ is equal to 1.