Let $T$ be a linear operator with rank $1$ on a finite dimensional vector space $V$.Then Which of the following are true?
1)either $T$ is diagonalizable or $T$ is nilpotent.
2)$T$ is both diagonalizable and nilpotent.
I take $T$ as constant mapping and got it as diagonalizable. So can we say that 1) is true and 2) is false? Is there any other method to solve the problem?