A diagonalizable matrix ?

The only eigenvalue of this matrix is 1 (with algebraic multiplicity 2). So if it was diagonalizable, it would be similar to the identity matrix :

$$\left(\begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\right)=P\left(\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right) P^{-1}=\left(\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right)$$which is a contradiction.