Question2.
Let A = [2 1 1
1 2 1
1 1 2 ].
(a) Find the characteristic polynomial PA(?) of A and theeigenvalues of A. For convenience, as usual, enumerate theeigenvalues in decreasing order ?1 ? ?2 ? ?3.
(b) For each eigenvalue ? of A find a basis of the correspondingeigenspace V (?). Determine (with a motivation) whether V (?) is aline or a plane through the origin. If some of the spaces V (?) isa plane find an equation of this plane.
(c) Find a basis of R 3 consisting of eigenvectors if such basisexist. (Explain why or why not). Is the matrix A diagonalizable? If”yes”, then write down a diagonalizing matrix P, and a diagonalmatrix ? such that A = P?P ?1 , P ?1AP = ?. Explain why the matrixP is invertible but do not compute P ?1 .
(d) Consider the eigenvalues ?1 > ?3. Is it true that theorthogonal complements of the eigenspaces satisfy (V?1 ) ? = V?3 ,(V?3 ) ? = V?1 ? Why or why not??
