<s>[INST] How find this matrix in an equation 
Let $x$ is give numbers.and define  the matrix $A$ such
  $$A^T\cdot\begin{bmatrix}
1&x\\
x&1
\end{bmatrix}\cdot A=1$$
this book say it is clear have
  $$A=\dfrac{1}{\sqrt{2(1+x)}}\begin{bmatrix}
1\\
1
\end{bmatrix}$$

I think this not true,let
$$A=\begin{bmatrix}
a\\
b
\end{bmatrix}$$
then we have
$$A^T\cdot\begin{bmatrix}
1&x\\
x&1
\end{bmatrix}\cdot A=\binom{a+bx}{b+ax}\cdot\begin{bmatrix}
a\\
b
\end{bmatrix}=a^2+b^2+2abx=1$$
First which is wrong?
why is "clear" can you explain? Thank you,If not it is clear,can you solve this equation
 [/INST] This is true and clear 
if you let $$A=\begin{bmatrix}
a\\
b
\end{bmatrix}$$
then we have
$$A^T\cdot\begin{bmatrix}
1&x\\
x&1
\end{bmatrix}\cdot A=\binom{a+bx}{b+ax}\cdot\begin{bmatrix}
a\\
b
\end{bmatrix}=a^2+b^2+2abx=1$$
the book assumed a case where $a=b$.
This will then give $a^2+a^2+2a^2x=1$ and solving for a gives:
$$a=\dfrac{1}{\sqrt{2(1+x)}}$$
Therefore we will have
$$A=\dfrac{1}{\sqrt{2(1+x)}}\begin{bmatrix}
1\\
1
\end{bmatrix}$$
</s>