Determine whether the given matrices are diagonalizable 1 4

Determine whether the given matrices are diagonalizable. [1 4 1 -2] [1 1 -2 4 0 4 1 -1 4]

Solution

a) characteristic polynomial:

eigenvalues = -3 and 2

eigenvectors

for Eigenvalue -3:
   [ -1 ; 1 ]

for Eigenvalue 2:
   [ 4 ; 1 ]

hence it is diagonalizable

b)

Characteristic Polynomial:
-x^3 + 5x^2 - 6x

Real Eigenvalues:   { 0 ; 2 ; 3 }

Eigenvectors:

for Eigenvalue 0:
   [ -1 ; 3 ; 1 ]

for Eigenvalue 2:
   [ 0 ; 2 ; 1 ]

for Eigenvalue 3:
   [ -1 ; 0 ; 1 ]

diagonalizable