let A be an nxn matric and the trace of A is defined by tr(A)= a_11+a_22...+a_nn . Prove that the coeff of x^(n-1) in the characteristic polynomial is -tr(A)

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- Dec 1st 2010, 08:39 AMalexandrabel90trace
let A be an nxn matric and the trace of A is defined by tr(A)= a_11+a_22...+a_nn . Prove that the coeff of x^(n-1) in the characteristic polynomial is -tr(A)

- Dec 1st 2010, 09:14 AMFernandoRevilla
We have

and the terms we haven't written have degree .

Regards.

Fernando Revilla - Dec 1st 2010, 02:27 PMDrexel28
Another method (the first is similar to that of Dr. Revilla's) is that you can first recall that is monic and it's roots are precisely the mulitiset spectrum of . and so

We can then recall Vieta's formula so that (remembering that is monic) that where is the first symmetric polynomial. Thus,