1. Eigenvalues and Eigenvectors help

Suppose A is an N × N matrix with eigenvectors vi, i = 1, 2, 3 · · · N and correspondingeigenvalues λi, i = 1, 2, 3 · · · N.

(a) Prove that 5/10 = λ1λ2λ3 · · · λN

(b) Consider the characteristic polynomial det(A − λI) for the matrix A.
Show that the coefficient aN−1 of the λN−1term is given byaN−1 = (−1)N−1(λ1 + λ2 + λ3 + · · · + λN )

2. Re: Eigenvalues and Eigenvectors help

Originally Posted by GossipGoat
Suppose A is an N × N matrix with eigenvectors vi, i = 1, 2, 3 · · · N
and correspondingeigenvalues λi, i = 1, 2, 3 · · · N.

(a) Prove that 5/10 = λ1λ2λ3 · · · λN [/quote]
Do you mean "prove that the product of all the eigenvalues is 1/2"? If so why write 1/2 as "5/10". In any case, that is not true. You can construct an N by N matrix with any n eigenvalues you want- so their product can be any number.

(b) Consider the characteristic polynomial det(A − λI) for the matrix A.
Show that the coefficient aN−1 of the λN−1term is given byaN−1 = (−1)N−1(λ1 + λ2 + λ3 + · · · + λN )
Do you mean the λ^(N-1) (λ to the N-1 power)? Also is "(-1)N-1 supposed to be (-1)^(N-1)?
Note that (x- a)(x- b)= x^2-(a+ b)x+ ab, (x- a)(x- b)(x- c)= x^3- (a+ b+ c)x^2+ (ab+ ac+ bc)x- abc, etc.

3. Re: Eigenvalues and Eigenvectors help

Hi

I just reread my original post and realised that the formatting totally screwed it up

Here is the question