the inverse function with 2 variables

Question

$f(x,y)=(x+y+1,x-y-1)$. Find the inverse of $f$.

$f^{-1}(x+y+1,x-y-1)=(x,y).$

Let $u=x+y+1$ and $v=x-y-1.$

Here, without $1$ and $-1$, I can invert this:

$$\begin{bmatrix}1&1\\1&-1\end{bmatrix}^{-1}=\frac 1{-2}\begin{bmatrix}-1&-1\\-1&1\end{bmatrix}.$$

Then, $f^{-1}(x,y)=(\frac{x+y}{2},\frac{x-y}{2}).$ But, I don't know how to deal with $1$ and $-1$. I appreciate any hint.

Answer 1

We have that

$$f(x,y)=\begin{bmatrix}f_1\\f_2\end{bmatrix}=\begin{bmatrix}1&1\\1&-1\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}+\begin{bmatrix}1\\-1\end{bmatrix}$$

then

$$\begin{bmatrix}1&1\\1&-1\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}f_1\\f_2\end{bmatrix}-\begin{bmatrix}1\\-1\end{bmatrix}$$

and thus

$$\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}1&1\\1&-1\end{bmatrix}^{-1}\begin{bmatrix}f_1\\f_2\end{bmatrix}-\begin{bmatrix}1&1\\1&-1\end{bmatrix}^{-1}\begin{bmatrix}1\\-1\end{bmatrix}$$