Say we have a matrix $A$ which is Diagonalizable, and $P$ which can be used in diagonalizing $A$. We are tasked to find the diagonal matrix $D$ such that $A = PDP^{-1}$
Since we know $A,P$, and it is given that $P$ can be used in diagonalizing $A$ why can't we just do $P^{-1}AP = D$?
$A = \begin{pmatrix} -3&-4\\ -9&-3\\ \end{pmatrix}, P = \begin{pmatrix} 2&-2\\ 3&3\\ \end{pmatrix}, P^{-1} = \begin{pmatrix} 3&2\\ -3&2\\ \end{pmatrix}$
Multiplying $P^{-1}AP$ yields $\begin{pmatrix} -108&0\\ 0&36\\ \end{pmatrix}$
Yet, $D$ is actually $\begin{pmatrix} -9&0\\ 0&3\\ \end{pmatrix}$
May someone explain to me why, if we know $A,P,P^{-1}$ and we use legitimate operations to compute $D$, we still don't get $D$? I mean I get a form of $D$ (aka. row equivalent, if I were to divide by $12$, but I don't know this just doing the calculation itself)..
Thanks in advance!
You're getting the wrong result because your inverse is incorrect. You are forgetting to divide by the determinant of $P$.