I'm struggling with this problem:
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 = det (Aii) where Aii is the (i,i) minor of A.
Similarly, what is the coefficient cr?
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.
What should I do?