For which value of $a$ is the matrix is isometric? 
For  which value  of $a$  is the matrix   $\begin{pmatrix} a & 0 \\1 &1\ \end{pmatrix}$ is isometrics   ?

My attempt : For isometric it must be  unitary so $a = -1$
Is its true ?

Suppose $\begin{pmatrix} a & 0 \\1 &1\ \end{pmatrix}\begin{pmatrix} x \\y\ \end{pmatrix}$ and  $\begin{pmatrix} x \\y\ \end{pmatrix}$ have the same length . 
Then $a^2x^2+(x+y)^2=x^2+y^2$ which is  not possible for all $x$ and $y$.
