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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.

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    @Nunoxic: "Over", not "in". The matrix elements are in $\mathbb R$.2012-02-12

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Hints:

1) Consider rotation matrices.

2) Show that the only possible eigenvalues are $1$ and $-1$.

3) What's the minimum value of $x^2 + x + 1$ for real $x$?

4) Do you mean $f$ is the characteristic polynomial of $A$? Then $\det((A-I)^2) = f(1)^2$.