max x'Ax subject to x'x = 1 for A a 2 by 2 matrix and x a 2 by 1 vector.

max x'Ax subject to x'x = 1

for A a 2 by 2 matrix and x a 2 by 1 vector. My advice is to write out x'Ax and x'x in full, set up the Lagrangian as you've been doing in micro, solve the problem for x_1 and x_2 as well as the Lagrangian multiplier, and then see if you can interpret your solution in terms of eigenvectors and eigenvalues.

By the way, this is the first step in developing the method of principal components, if you've heard of that.