Matrix of the operator $A(x,y)=(y-x+iy,x-2y-ix)$

Question

I got an operator $A: \Bbb C^2 \to \Bbb C^2$ given as $A(x,y)=(y-x+iy,x-2y-ix)$

and I want to represent it as a matrix, so I could find then the orthonormal basis to which would have this operator a diagonal matrix form.

But I am not sure how to put it into the matrix, it confuses me:

$$ A= \begin{pmatrix} -1 & 1 -i \\ 1+i & -2 \\ \end{pmatrix} $$ or

$$ A= \begin{pmatrix} -1 & 1 +i \\ 1-i & -2 \\ \end{pmatrix} $$

Answer 1

Since $A(1,0)=(-1,1-i)$ and $A(0,1)=(1+i,-2)$, the second option is the correct one.