# For each real numbers $x$ such that $-1 < x < 1$ let $A(x)$ be the matrix.$(1-x)^{-\large\frac{1}{2}}\begin{bmatrix}1&-x\\-x&1\end{bmatrix}$ and $z=\large\frac{x+y}{1+xy}$.Then
$\begin{array}{1 1}(A)\;A(z)=A(x)+A(y)&(B)\;A(z)=A(x)[A(y)]^{-1}\\(C)\;A(z)=A(x)A(y)+\sqrt{1+xy}&(D)\;A(z)=A(x)-A(y)\end{array}$