Diagonalizable matrix with only one eigenvalue

Diagonalizable matrix with only one eigenvalue

I have a question from a test I solved (without that question.. =) "If a
matrix A s.t A is in M(C) (in the complex space) have only 1 eigenvalue
than A is a diagonalizable matrix"
That is a false assumption since a (nXn matrix) a square matrix needs to
have at least n different eigenvalues (to make eigenvectors from) - but
doesn't the identity matrix have only 1 eigenvalue?...