Matrix $A$ has an eigenvalue with multiplicity $>1$, is $A$ diagonalisable?

Question

Matrix $A$ has an eigenvalue with multiplicity $>1$, is $A$ diagonalisable?

I know that if $A$ has distinct eigenvalues $\Rightarrow$ all eigenvectors are linearly independent $\Rightarrow$ I can find an inverable $P$ s.t. $P^{-1}AP=D$ where $D$ is the diagonal matrix with my distinct eigenvalues

Answer 1

No. For example

$$\begin{pmatrix}1&1\\0&1\end{pmatrix}$$ has the eigenvalue with multiplicity $2$ but it isn't diagonalizable.

Notice A matrix is diagonalizable if and only if the multiplicity of every eigenvalue equals to the dimension of the corresponding eigenspace.