# If two matrices have the same characteristic polynomials, determinant and trace, are they similar?

If two $n \times n$ matrices have the same characteristic polynomials, determinant and trace, are they similar, EVEN if ($\lnot \#Spec= 0$)?

The answer is no. Take the identity matrix and the matrix $$a_{11}=a_{12}=a_{22}=1, a_{21}=0.$$