connection between trace and determinant to matrices similarity in 2X2 matrices

Question

i'm struggling explaing these statments

Let A,B be matrices of order 2X2 over R

1. given det(A)=det(B)=4 and tr(A)=tr(B)=7 A is similar to B ( should be True)
2. given det(A)=det(B)=4 and tr(A)=tr(B)=4 A is similar to B ( should be false)

i found that the charcristic polynomiala of the two statments are (1) t^2 -7t + 4
(2) t^2 -4t + 4 = (t-2)^2

over R, (1) is not Diagonalizable, and (2) is , what am i missing ? how can i determine these statments ?

what is the connection between trace and determinant here ?

Answer 1

1. If $\alpha$ and $\beta$ are the eigenvalues of these matrices, then $\alpha\ne\beta$. So, $A$ and $B$ are similar to $\left[\begin{smallmatrix}\alpha&0\\0&\beta\end{smallmatrix}\right]$.
2. Consider $\left[\begin{smallmatrix}2&0\\0&2\end{smallmatrix}\right]$ and $\left[\begin{smallmatrix}2&1\\0&2\end{smallmatrix}\right]$, which are not similar. However both have determinant equal to $4$ and trace also equal to $4$.