Give an example of two 4x4 matrices which have the same char

Give an example of two 4x4 matrices which have the same characteristic and minimal polynomials. but which are not similar.

Show that this cannot happen for 3x3 matrices, i.e. if A, B are 3x3 matrics with the same characteristic and minimal polynomials. then A, B are similar

Solution

M = [ 1 1 0 0 ]

[ 0 1 0 0 ]

[ 0 0 1 1 ]

[0 0 0 1]

N = [1 0 0 0]

[0 1 0 0 ]

[0 0 1 1]

[0 0 0 1]