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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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    $\begingroup$ "Unitary" is the correct adjective for matrices (Unitarian is a description of a religious belief). Is $R$ supposed to be $\mathbb{R}$? Note that matrices $M$ such that $\lVert M\mathbf{x}\rVert = \lVert \mathbf{x}\rVert$ for all $\mathbf{x}$ are called "unitary" when working over the complex numbers, but are called orthogonal when working over the real numbers. $\endgroup$ – Arturo Magidin Feb 10 '12 at 20:08
  • $\begingroup$ How can one have matrices in $\mathbb{R}$? $\endgroup$ – Inquest Feb 11 '12 at 6:47
  • $\begingroup$ @Nunoxic: "Over", not "in". The matrix elements are in $\mathbb R$. $\endgroup$ – Robert Israel Feb 12 '12 at 6:06
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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 $\det((A-I)^2) = f(1)^2$.

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