/sci/ - Science & Math

Hey guys I've been struggling with this problem for a while now, and I need some help

A real [math]n \times n[/math] matrix [math]A[/math] has an eigenvalue [math]\lambda[/math]. Show that there exists some [math]k \in \mathbb{Z}^{+}[/math] with [math]k \le n[/math] such that
[eqn]
|\lambda - A_{kk}| \le \sum_{j=1, j \not= k}^{n}|A_{jk}|.
[/eqn]

So I thought of assuming that [math]|\lambda - A_{kk}| > \sum_{j=1, j \not= k}^{n}|A_{jk}| \:\: \forall \:\: k \le n [/math] and proving it by contradiction, but I don't seem to be getting anywhere. It was very easy to prove for the specific case of [math]n=2[/math] but I can't even do it for [math]n=3[/math].
I feel like this must have something to do with algebraically manipulating [math]\det(A - \lambda I)[/math] but I really have no clue what to do.