In this lecture we study eigenvectors and eigenvalues of an n x n matrix A (motivated by the search for lines through the origin left fixed by the transformation corresponding to A). We study examples where the characteristic polynomial has distinct roots (i.e. distinct eigenvalues) and A is diagonalisable. We also study examples where the characteristic polynomial has a repeated root, but in some cases A is diagonalisable and in others A is not. We use our results to calculate a high power of a matrix.