If A is an nxn matrix, then A has n distinct eigenvalues.True or false?

The statement is false. As a counterexample, let 
$$A=\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}=I. $$
Let $\lambda$ be an eigenvalue. Then 
\[\text{det}(A-\lambda I)=\text{det}\begin{pmatrix}1-\lambda & 0 \\ 0 & 1-\lambda \end{pmatrix} \]
\[=(1-\lambda)^2=0.\]
Hence
\[\lambda=1,1.\]
So there is only one eigenvalue and NOT two distinct eigenvalues.