Diagonalize the matrix using simultaneous row and column operations?

Question

I need to diagonalize the matrix $$\left[\matrix{1 & -1\\-1 &3}\right]$$ using simultaneous row and column operations.

My attempt is to do $R2+R1 \to R2$ and $C2+C1\to C2$, and this brings the matrix to $\left[\matrix{1 & 0\\0 &-4}\right]$. Doing the same row operations to the identity matrix, I get $\left[\matrix{1 & 0\\1 &1}\right]$.

From what I understand, in the form $P^TAP = D$, is that $P^T=\left[\matrix{1 & 0\\1 &1}\right]$ and $D=\left[\matrix{1 & 0\\0 &-4}\right]$.

But I have plugged this into Wolfram Alpha and I know that my answer is incorrect. I'm not sure where to go from here.

Answer 1

When I calculate $R2+R1 \to R2$ and $C2+C1\to C2$, I get $$ \left[\matrix{1 & 0\\0 &2}\right]. $$ Check out your calculations once again. Note $\color{green}{2}=\color{red}{+}3-1$, $\color{green}{-4}=\color{red}{-}3-1$.

Answer 2

That matrix does not have eigenvalues 1 and 2 for sure. Also if you want to preserve similarity it is necessary that the simultaneous column operation is the inverse of the raw operation just done. Which is not in your case. You do get a diagonal matrix, but is not similar to the one you started with.