If possible, I have to find an invertible matrix $P$ and a diagonal matrix $D$ such that $A=PDP^{-1}$. $A = \begin{pmatrix} -9 & 8 & -8\\ -4 & 3 & -4\\ 2 & -2 & 1 \end{pmatrix}$ it's eigen values are 3 with multiplicity 1, and -1 with multiplicity 2
$A-3I_3 = \begin{pmatrix} -12 & 8 & -8\\ -4 & 0 & -4\\ 2 & -2 & -2 \end{pmatrix}$
For the reduced row echelon from of $A-3I_3$ I get the identity matrix $I_3$ and I would get the same thing for the reduced row echelon form of $A+I_3$ which would mean this matrix is not diagonalizable
I'm not sure what I am doing wrong (might be a minor arithmetic error I just can't identify) because when I look up the answer it says it is diagonalizable with $P =\begin{pmatrix} -4 & -1 & 1\\ -2 & 0 & 1\\ 1 & 1 & 0 \end{pmatrix}$ and $D = \begin{pmatrix} -3 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & -1 \end{pmatrix}$
Any help would be appreciated.