let's say you are given the following matrix(adsbygoogle = window.adsbygoogle || []).push({}); T:

T_{11}T_{12}. . . T_{1n}

T_{21}T_{22}. . . T_{2n}

. . .

. . .

. . .

T_{n1}T_{n2}. . . T_{nn}

You want to find the characteristic equation for the matrix, which is

[itex]C_{n} \lambda^{n} + C_{n-1} \lambda^{n-1} + ... + C_{1} \lambda + C_{0} = 0.[/itex]

Now, [itex]C_{n} = (-1)^{n}[/itex], [itex]C_{n-1} = (-1)^{n-1} Tr(T)[/itex], and [itex]C_{0} = det(T)[/itex].

I have been having a pretty hard proving these three formulae. The first one looks intuitively obvious and I think I have a fairly good understanding of why the second one has the structure that it has. But the complexity of the matrix notation and the fact that there are so many elements in the matrix is halting my attempts to precisely determine each of the formulae from the matrix. I would appreciate any help. Thanks.

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# Coefficients of the characteristic equation

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