(*Exercise 1 a) *)
A={{3,4},{4,3}};
A//MatrixForm

(*Computations done on the board gave us the following eigenvectors*)
P={{1,1},{-1,1}};
P//MatrixForm

diag=Inverse[P].A.P//MatrixForm

(*Checking*)
Eigensystem[A]

(*Exercise 1 b) *)
A={{1,0,0},{1,0,0},{1,2,4}};
A//MatrixForm

Eigensystem[A]

(*Exercise 1 c) *)
A={{0,0,0},{1,0,0},{1,0,0}};
A//MatrixForm

Eigensystem[A]