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For a given $n \times n$-matrix $A$, and $J\subseteq\{1,...,n\}$ let us denote by $A[J]$ its principal minor formed by the columns and rows with indices from $J$.

If the characteristic polynomial of $A$ is $x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$, then why $$a_k=(-1)^{n-k}\sum_{|J|=n-k}A[J],$$ that is, why is each coefficient the sum of the appropriately sized principal minors of $A$?

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    Found something useful .. www.mcs.csueastbay.edu/~malek/Class/Characteristic.pdf2012-05-12
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    Also http://books.google.co.in/books?id=ULMmheb26ZcC&pg=PA195&dq=coefficients+of+characteristic+polynomial&hl=en&sa=X&ei=q9utT7PpFI3JrAeWpcWUBA&redir_esc=y#v=onepage&q=coefficients%20of%20characteristic%20polynomial&f=false2012-05-12
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    www.maa.org/sites/default/files/Louis_L49930._Pennisi.pdf2016-05-16
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    This follows from Corollary 5.161 in my [*Notes on the combinatorial fundamentals of algebra*, version of 25 May 2017](https://github.com/darijgr/detnotes/releases/tag/2017-05-25). Just mentioning this for the sake of completeness; I'm sure you don't want to read my proof (which is an unenlightening orgy of notation, with nothing interesting going on other than repeated applications of multilinearity), but it might be comforting to know it exists.2017-07-18
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    See also https://math.stackexchange.com/a/336078/ for an outline of the proof.2017-07-18

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