well, I know that the trace is the negative of coefficient of $x^{226}$ of the characteristic polynomial the matrix, but I dont know how the Char.Poly looks like in this case.please give me some hint. Do I have to work in the splitting field of the characteristic polynomial and add the eigen values to get the trace?but I dont know how.
Trace of a $227\times 227$ matrix over $\mathbb{Z}_{227}$
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linear-algebra
abstract-algebra
matrices
1 Answers
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Let's do something smaller. Use 3 instead of 227. The matrix $\pmatrix{0&0&0\cr0&1&0\cr0&0&2\cr}$ has distinct eigenvalues and trace zero. The matrix $\pmatrix{1&0&0\cr0&0&1\cr0&2&0\cr}$ has distinct eigenvalues $1,i,-i$, where $i$ is a square root of minus one in an extension field, and it has trace 1. So if the eigenvalues are allowed to be in an extension field, the answer is not determined by the information given.
Of course, if the eigenvalues have to be in the field, the problem is easy.
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0yes thank you . – 2012-07-22