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evinda

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I want to prove that if a $m \times m$ matrix $A$ has rank $m$ and satisfies the condition $A^2=A$ then it will be identical.

$A^2=A \Rightarrow A^2-A=0 \Rightarrow A(A-I)=0$.

From this we get that either $A=0$ or $A=I$.

Since $A$ has rank $m$, it follows that it has $m$ non-zero rows, and so it cannot be $0$.

Thus $A=I$. Is everything right? Or could something be improved? (Thinking)