Consider the matrix

$A=\left[\begin{array}{ccc}2& -2& 1\\ -1& 3& -1\\ 2& -4& 3\end{array}\right]$

$($a$)$ Write the definition of an eigenvalue and eigenvector of a matrix.

$($b$)$ Find the eigenvalues and eigenvectors of matrix A$.$

$($c$)$ If possible write a basis for $R^3$ consisting entirely of eigenvectors of $A.$