Please see the attached Word document.
Thank-you for your help.
Suppose that A is a square matrix, and that the only eigenvectors of A are scalar multiples of the vector:
Deduce as much information as you can about A, by answering the following questions. Give reasons for your answers.
(a) What size is A?
(b) How many eigenvalues does A have? Can you tell what the eigenvalues are?
(c) What can you tell about the algebraic and geometric multiplicities?
(d) Can you tell whether or not A is diagonalizable?
This provides examples of working with a matrix regarding eigenvalues, diagonalizability, and multiplicities.