Matrix/Trigonalizable/Not diagonalizable/Not bijective/Nilpotent/Exercise

a) Let ${}M$ be a ${}2\times 2$-matrix that is trigonalizable, but neither diagonalizable nor invertible. Show that ${}M$ is nilpotent.

b) Give an example of a ${}3\times 3$-matrix ${}M$ that is trigonalizable, but neither diagonalizable, nor invertible, nor nilpotent.

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