Diagonalization of a given matrix

Let $A = [\begin{matrix} 2 & - 1 & - 1 \\ 1 & 0 & - 1 \\ 1 & - 1 & 0 \end{matrix}]$ .

Compute the characteristic polynomial $p_{A} (x)$ of $A$ . It has integer roots.
For each eigenvalue $λ$ of $A$ , find a basis for the eigenspace $E_{λ}$ .
Determine if $A$ is diagonalizable. If so, give matrices $P$ and $B$ such that $P^{- 1} A P = B$ and $B$ is diagonal. If no, explain carefully why $A$ is not diagonalizable.

View

L A T E X

code

Let $A=\begin{bmatrix} 2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0\end{bmatrix}$.
\begin{enumerate}[label=\alph*)]
	\item Compute the characteristic polynomial $p_A(x)$ of $A$. It has integer roots.
	\item For each eigenvalue $\lambda$ of $A$, find a basis for the eigenspace $E_{\lambda}$.
	\item Determine if $A$ is diagonalizable. If so, give matrices $P$ and $B$ such that $P^{-1}AP=B$ and $B$ is diagonal. If no, explain carefully why $A$ is not diagonalizable.
\end{enumerate}