A is a 5x5 matrix with rank 3. Which two of the following conditions are required for the matrix to be diagonalizable? (There may be multiple correct answers.)
(I) nullity(A-2I) = 2
(II) nullity(A-3I) = 3
(III) rank(A-2I) = 2
(IV) rank(A-3I) = 3
I believe that nullity is just the number of free variables and rank is just the number of basic variables. I know that a matrix of size n x n needs n linearly independent eigenvectors to be diagonalizable (or n distinct eigenvalues which guarantees n l.i. eigenvectors). I also know that nullity(A-yI) = multiplicity of y for a diagonalizable matrix. I just don't get how to find the multiplicity, or how else to approach the question (if multiplicity is the wrong approach).