There is $2 \times 2$ real matrix with characteristic polynomial $x ^ { 2 } + 1$.

Let $A \in \mathcal{M}_2(\mathbb{R})$. Compare the eigenvalue circle of $A$ and that of its transpose.

Let $A \in \mathcal{M}_2(\mathbb{R})$. Determine a necessary and sufficient condition on $\mathcal{CP}_A$ for $A$ to be symmetric.

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$ and $(x,y,z)$ in $\mathbb{R}^3$, we denote by $\psi_A(x,y,z)$ the real part of the determinant of the matrix $\left(\begin{array}{cc} a-x-\mathrm{i}z & b-y \\ c+y & d-x-\mathrm{i}z \end{array}\right)$, where $\mathrm{i}$ is the complex affix of the point $J = (0,1)$.
Calculate $\psi_A(x,y,z)$.

The purpose of this question is to show that $\sqrt [ 3 ] { 2 }$ is not an eigenvalue of a symmetric matrix with coefficients in $\mathbb { Q }$. We reason by contradiction, assuming the existence of a matrix $M \in S _ { n } ( \mathbb { Q } )$ (for some integer $n$) for which $\sqrt [ 3 ] { 2 }$ is an eigenvalue.
4a. Show that $X ^ { 3 } - 2$ divides the characteristic polynomial of $M$. (One may begin by proving that $\sqrt [ 3 ] { 2 } \notin \mathbb { Q }$.)
4b. Conclude.

The purpose of this question is to show that $\sqrt [ 3 ] { 2 }$ is not an eigenvalue of a symmetric matrix with coefficients in $\mathbb { Q }$. We reason by contradiction, assuming the existence of a matrix $M \in S _ { n } ( \mathbb { Q } )$ (for some integer $n$) for which $\sqrt [ 3 ] { 2 }$ is an eigenvalue.
4a. Show that $X ^ { 3 } - 2$ divides the characteristic polynomial of $M$. (One may begin by proving that $\sqrt [ 3 ] { 2 } \notin \mathbb { Q }$.)
4b. Conclude.

Let $G _ { 1 } = \left( S _ { 1 } , A _ { 1 } \right)$ and $G _ { 2 } = \left( S _ { 2 } , A _ { 2 } \right)$ be two non-empty graphs such that $S _ { 1 }$ and $S _ { 2 }$ are disjoint, that is, such that $S _ { 1 } \cap S _ { 2 } = \varnothing$. Let $s _ { 1 } \in S _ { 1 }$ and let $s _ { 2 } \in S _ { 2 }$.
We define the graph $G = ( S , A )$ with $S = S _ { 1 } \cup S _ { 2 }$ and $A = A _ { 1 } \cup A _ { 2 } \cup \left\{ \left\{ s _ { 1 } , s _ { 2 } \right\} \right\}$.
Show that : $$\chi _ { G } = \chi _ { G _ { 1 } } \times \chi _ { G _ { 2 } } - \chi _ { G _ { 1 } \backslash s _ { 1 } } \times \chi _ { G _ { 2 } \backslash s _ { 2 } }$$