Surprise.. stuck on yet another hw problem Are these two matrices similar?
Well, the simplest check is usually the determinant. In this case, it doesn't help since both matrices have determinant equal to 16.
On the other hand, you can look at the characteristic equation of each matrix (The equation with eigenvalues as roots). The first matrix gives you $\displaystyle (\lambda-2)(\lambda-2)(\lambda-4)$ while the second gives you $\displaystyle (\lambda-1)(\lambda-4)(\lambda-4)$. Since they are not the same, these two matrices are not similar!
yes, but...
what you said is true, however, two matrices with the same eigenvalues may not be similar. tests (in increasing order of strength):
determinants < eigenvalues < characteristic polynomials < Jordan forms
if any of these things are different for 2 matrices, we can conclude the matrices are NOT similar. but for any of these things except the jordan form, it might be the case that they have the same (determinant, eigenvalues, char. polyn.) but are not similar.