Uncorrelated estimators means the off diagonal of x transpose x itself vanishes

Consider the linear model $Y_{1} = \beta_{1}+\beta_{2}+\varepsilon_{1}, Y_{2} = \beta_{1}+2\beta_{2}+\varepsilon_{2}, Y_{3} = \beta_{1}+c\beta_{2}+\varepsilon_{3}$, where $\beta_{1},\beta_{2} \in \mathbb{R}$ are unknown parameters, and the uncorrelated errors $\varepsilon_{i}, i=1,2,3$ have zero mean and finite variance $\sigma^{2}(>0)$. The constant c is such that $\beta$̂$_{1}$ and $\beta$̂$_{2}$ are uncorrelated, where $\beta$̂$_{1}$ and $\beta$̂$_{2}$ are the best linear unbiased estimators of $\beta_{1}$ and $\beta_{2}$, respectively. Which of the following statements is the correct option for $(Var(\beta$̂$_{1}), Var(\beta$̂$_{2}))$?