Is an $n \times n$ matrix with rank $n$ always diagonalizable?

Question

I have an assignment problem:

Which statement(s) about an $n \times n$ matrix $A$ is/are always true?

1. If the rank of $A$ is $n$, then $A$ is diagonalizable.

2. If $A$ has $n$ distinct eigenvalues, then $A$ is diagonalizable.

3. If one or more of the eigenvalues of $A$ is $0$, then $A$ cannot be diagonalized.

I know that $\#2$ is true, because if it has $n$ distinct eigenvalues, then it has $n$ linearly independent eigenvectors, then it is diagonalizable.

I know that $\#3$ is false, for example the matrix of all zeroes is diagonalizable (since it's already diagonal) and has zero eigenvalues, so $\#3$ is clearly false.

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I am not sure about $\#1$. I know that rank $n$ means that the columns are all linearly independent. I think it means that it is diagonalizable but I am not certain.

Thanks for help.