Expanding $\det(\lambda I-A)$ gives $\lambda^3-17\lambda^2+43\lambda-27=(\lambda-1)(\lambda^2-16\lambda+27)$. The quadratic roots are $(16\pm\sqrt{256-108})/2=8\pm\sqrt{37}$. Thus **C**. Their sum is 17,the trace, and their product is 27,the determinant.
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