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If $A,B\in M_{4}\left( \mathbb{R} \right) :A\neq B;Tr\left( A\right) \neq 0$ and

$$\left\{\begin{matrix} A^2 - 2B + I_4 = 0_4 \\ B^2 - 2A + I_4 = 0_4 \end{matrix}\right.$$

prove that: $$A+B=-2I_{4}$$ and $$\det\left( A-aI_{4}\right) \geq \det\left( A+aI_{4}\right) ,\forall a\in \mathbb{R} $$

All my ideas ($A^{2}=2B-I_{4};B^{2}=2A-I_{4}\Rightarrow A^{2}B=2B^{2}-B;B^{2}A=2A^{2}-A$) seem to be wrong as long as I can't rich a prove.

R.W
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Brain123
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  • Suppose we look at $A$ and $B$ as complex matrices. If $v$ is a complex eigenvector of $A$ with eigenvalue $\lambda$. What do the equations say about $Bv$? And do we get a relation for $\lambda$ by relating the two equations? – Bob Jones Mar 16 '17 at 18:37
  • I can't understand this, Could you give me further Information about this? – Brain123 Mar 17 '17 at 09:00
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    Suppose $Av=\lambda v$. Apply both sides of the first equation to $\lambda$. What does it tell you $Bv$ is? – Bob Jones Mar 17 '17 at 09:10

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