# Coefficients of characteristic polynomial (linear algebra)

 P: 146 1. The problem statement, all variables and given/known data A is an nxn square, real matrix. Let f(x) be the characteristic polynomial, write f(x) = xn - c1xn-1 + ... + (-1)rcrxn-r + ... + (-1)ncn Show that cn-1 = $$\sum$$ det (Aii) where Aii is the (i,i) minor of A. Similarly, what is the coefficient cr? 2. Relevant equations 3. The attempt at a solution I have shown that c1 = trace(A) and cn = det(A). cn-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 Aii? If this is true, how would I express it more rigorously? Also I'm not sure how to generalise for cr.