Invariance of estimable functions

Let $\lambda'\beta$ be an estimable function of $\beta$ in the model $\mathbf Y=X\beta+\varepsilon$, where $\mathbb{E}(\mathbf Y)=X\beta$ and $X$ is an $n\times p$ matrix of rank $k<p\le n$. Let $\hat\beta$ be any solution to the normal equations $X'X\hat\beta=X'\mathbf Y$ and let $\mathbf r$ be any solution to $X'X\mathbf r=\lambda$. Consider the following statements for the two estimators $\lambda'\hat\beta$ and $\mathbf rX'\mathbf Y$:

I. $\lambda'\hat\beta$ is equal to $\mathbf r'X'\mathbf Y$ for any $\hat\beta$ or $\mathbf r$.

II. $\lambda'\hat\beta$ and $\mathbf r'X'\mathbf Y$ are invariant to the choice of any $\hat\beta$ or $\mathbf r$.

Which of the statements given above is/are correct?