Given $A$ and $B$, $2\times 2$ matrices, which of the following is necessarily true?
If $A$ and $B$ are both Unitary matrices over $R$ and $\det(A)=\det(B)=1$ then $A$ is similar to $B$.
If $A$ and $B$ are both Unitary matrices over $R$ and $\det(A)=\det(B)=-1$ then $A$ similar to $B$.
If $A$ is a hermitian matrix and $B=A^2+A+I$, then $B$ is an invertible matrix
If $B=A^2-2A+I$ and the characteristic polynomial of $A$ is $f(x)=x^2-x$ then $\det(A)\neq \det(B)$.
Now I know the answers to this question but want a good explanation.
Thanks.