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Consider an $n \times n$ matrix $A=I_{n}-\alpha \alpha^{T}$, where $I_{n}$ is the identity matrix of order $n$ and $\alpha$ is an $n \times 1$ column vector such that $\alpha^{T} \alpha=1$. Prove that $A^{2}=A.$
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We just have to use the given condition for alpha

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