Eigenvalues of a 2×2 matrix

The matrix $\begin{pmatrix} -5 & 2 \\ -7 & 4 \end{pmatrix}$ has two eigenvalues $a$ and $b$ . Compute $a^2 + b^2$ .

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Eigenvalues solve $\lambda^2-(\operatorname{tr})\lambda+\det=0$ ; here $\operatorname{tr}=-1$ , $\det=-6$ .
$\lambda=2,-3$ ; or use $a^2+b^2=(\operatorname{tr})^2-2\det=1+12$ .

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