Is This Matrix Diagonalizable? Consider the matrix A below:
$$A = \begin{pmatrix} 3 & 2 & 2 \\ -1 & 0 & -2 \\ 1 & 2 & 4\end{pmatrix}$$
Is the matrix A diagonalizable? If so, then find a diagonal matrix D and an invertible matrix P such that $P^{-1}AP=D$.
I know it is supposed to be diagonalizable but I have tried to solve it and haven't succeeded. The format I'm used to is $A=PDP^{-1}$.
Thanks.

Doing some regular computations for eigenvalues and eigenvectors you should find
$$\matrix{
   {{\lambda _1} = 2} \hfill &  \to  \hfill & {{X_1} = \left[ {\matrix{
   { - 2}  \cr 
   0  \cr 
   1  \cr 
 } } \right]} \hfill & {{X_2} = \left[ {\matrix{
   { - 2}  \cr 
   1  \cr 
   0  \cr 
 } } \right]} \hfill  \cr 
   {{\lambda _2} = 3} \hfill &  \to  \hfill & {{X_3} = \left[ {\matrix{
   1  \cr 
   { - 1}  \cr 
   1  \cr 
 } } \right]} \hfill & {} \hfill  \cr
 } $$
and hence your matrix $P$ will be
$$P = \left[ {\matrix{
   {{X_1}} & {{X_2}} & {{X_3}}  \cr 
 } } \right] = \left[ {\matrix{
   \hfill { - 2} & \hfill { - 2} & \hfill 1 \cr 
   \hfill 0 & \hfill 1 & \hfill { - 1} \cr 
   \hfill 1 & \hfill 0 & \hfill 1 \cr 
 } } \right]$$
and finally
$$D = {P^{ - 1}}AP = \left[ {\matrix{
   2 & 0 & 0  \cr 
   0 & 2 & 0  \cr 
   0 & 0 & 3  \cr 
 } } \right]$$
will be the diagonal matrix where its diagonal elements are the eigenvalues. However, as $\lambda_1=2$ had multiplicity $2$, it will appear twice on the diagonal here! :)
