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All but one of the following conditions are equivalent to the $n\times n$ matrix $A$ being invertible.

Which one ISN'T?

The column vectors of $A$ span $\mathbb{R}^n$

The row vectors of $A$ span $\mathbb{R}^n$

$A^T$ is invertible

$A^2=A$

$A-I_n$ is nilpotent