Coefficients of characteristic polynomial (linear algebra)

[Kate2010]

Homework Statement

A is an nxn square, real matrix. Let f(x) be the characteristic polynomial, write f(x) = xⁿ - c₁x^n-1 + ... + (-1)^rc_rx^n-r + ... + (-1)ⁿc_n

Show that c_n-1 = \sum det (A_ii) where A_ii is the (i,i) minor of A.

Similarly, what is the coefficient c_r?

Homework Equations

The Attempt at a Solution

I have shown that c₁ = trace(A) and c_n = det(A).
c_n-1 is the coefficient of x, so is the sum of all products involving one entry from the diagonal, would this product then be the determinant of the matrix formed by deleting the row and column that this entry is in, so A_ii?

If this is true, how would I express it more rigorously?

Also I'm not sure how to generalise for c_r.

 

[Count Iblis]

Write det(A -x I) in the "sum over permutations" form. Then consider the form of the permuations that will get you a factor x from some specific position on the diagonal. You then consider that restricted sum over permutations and see if it corresponds to an unrestricted sum over permutations of a smaller set of rows and is thus, by definition, the determinant of a minor of the matrix.