The matrix $$A=\left[\begin{array}{ccc} -1 &1 &2\cr 0 &-2 &-2\cr 0 &1 &1 \end{array}\right]$$ has two real eigenvalues, one of multiplicity $1$ and one of multiplicity $2$. Find the eigenvalues and a basis for each eigenspace.

The eigenvalue $\lambda_1$ is and a basis for its associated eigenspace is $\Bigg\lbrace$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$ $\Bigg\rbrace.$

The eigenvalue $\lambda_2$ is and a basis for its associated eigenspace is $\Bigg\lbrace$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$, $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$ $\Bigg\rbrace.$