I'm pretty sure you have to use the dimension of the column space but I can't figure this out:
If $A$ is a $3\times 3$ matrix such that $A^2 = 0$, then show that the rank of $A = 1$.
If anyone can help, thanks!
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More generally, if an $n \times n$ matrix has rank $r$, that means that the column space of $A$ has dimension $r$, and the null space has dimension $n-r$. In order for $A^2 = 0$ it is necessary and sufficient that the column space is contained in the null space, and this can only happen if $r \le n-r$, i.e. $r \le n/2$.
As stated, the conclusion does not follow: if $A=0$, then $A^2=0$, but $A$ does not have rank $1$, it has rank $0$.
If we add the assumption that $A$ is not the zero matrix, on the other hand, then the result does follow: