Let $A$ be an $n\times n$ matrix with entries in an arbitrary field $k$.
Is the characteristic polynomial $\det(tI_n-A)$ dependent only on the trace and determinant of $A$?
[Math] Characteristic polynomial depends only on the trace and determinant of the matrix
linear algebra
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Best Answer
No, look at $\left[\matrix{1&0&0\cr0&1&0\cr0&0&0 }\right]$ and $\left[\matrix{2&0&0\cr0&0&0\cr0&0&0 }\right]$. Both have determinant 0 and trace 2 but they do not have the same eigenvalues. Similar examples show it's not true for any $n>3$.