Question: Prove that the matrices $mathbf{H}$ and $mathbf{I}-mathbf{H}$ are idempotent, that is, $mathbf{H H}=mathbf{H}$ and $(mathbf{I}-mathbf{H})(mathbf{I}-mathbf{H})=mathbf{I}-mathbf{H}$.

Prove that the matrices $\mathbf{H}$ and $\mathbf{I}-\mathbf{H}$ are idempotent, that is, $\mathbf{H H}=\mathbf{H}$ and $(\mathbf{I}-\mathbf{H})(\mathbf{I}-\mathbf{H})=\mathbf{I}-\mathbf{H}$.

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