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I am trying to understand this derivation:

enter image description here

I think everything except the last equality is fairly simple, but I do not understand the last equality. Is there an error here?

I appreciate any help.

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There's no error.

Start with $X^\top X = (X^\top X+\lambda I) - \lambda I$; premultiply both sides by $(X^\top X+\lambda I)^{-1}$. Simplify.

Postmultiply by $\beta$

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