Sum of squared eigenvalues of a 2x2 matrix

Consider $A=\begin{bmatrix}-5 & 2\\ -7 & 4\end{bmatrix}.$ Call its two eigenvalues $a$ and $b$. Report $a^2+b^2$.

$13$. The characteristic polynomial is $\lambda^2+\lambda-6=(\lambda+3)(\lambda-2)$, giving eigenvalues $-3$ and $2$, so $9+4=13$. (Equivalently $a^2+b^2=\operatorname{tr}(A)^2-2\det A=1+12$.)