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Let $A$ be a $2\times 2$ complex matrix such that $A^2=0$. Prove that either $A=0$ or $A$ is similar over $\mathbb{C}$ to $$\left(\begin{array}{cc} 0 & 0 \\1 & 0 \end{array}\right) $$

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    Note that A is not invertible as its determinant is zero. So either A has rank 0 or 1. In the former case it is itself 0 and in the latter case it is similar to your matrix above.2012-09-16
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    I started with finding out the characterstic values and characteristic vectors.2012-09-22

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