For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$ and $(x,y)$ in $\mathbb{R}^2$, we denote by $\varphi_A(x,y)$ the determinant of the matrix $A(x,y) = \left(\begin{array}{cc} a-x & b-y \\ c+y & d-x \end{array}\right)$ and we consider $\mathcal{CP}_A$ the curve in $\mathbb{R}^2$ defined by the equation: $\varphi_A(x,y) = 0$. Verify that $\mathcal{CP}_A$ is a circle (we agree that a circle can be reduced to a point); we will call $\mathcal{CP}_A$ the eigenvalue circle of $A$. Specify its center $C_A$ and its radius $r_A$.
For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$ and $(x,y)$ in $\mathbb{R}^2$, we denote by $\varphi_A(x,y)$ the determinant of the matrix $A(x,y) = \left(\begin{array}{cc} a-x & b-y \\ c+y & d-x \end{array}\right)$ and we consider $\mathcal{CP}_A$ the curve in $\mathbb{R}^2$ defined by the equation: $\varphi_A(x,y) = 0$.
Verify that $\mathcal{CP}_A$ is a circle (we agree that a circle can be reduced to a point); we will call $\mathcal{CP}_A$ the eigenvalue circle of $A$. Specify its center $C_A$ and its radius $r_A$.