true or false diagonalisation

Question

For each of the following statements, select True or False

(i) Every square matrix is diagonalisable

(ii) If an  matrix is diagonalisable then it must have n distinct eigenvalues (i.e. all of them are different)

(iii) If a matrix A

 is diagonalisable with P^-1 AP=D then the diagonal matrix D has the eigenvaues of  A as the diagonal entries.

Answer 1

(i). FALSE An nxn matrix is diagonalizable if and only if n linearly independent eigenvalues exist.

(ii). FALSE An nxn matrix may have less than n distinct eigenvalues.

(iii) TRUE The eigenvalues are the entries on the main diagonal of the diagonal matrix (there may be repetitions)