The Same Temperature, Recorded Twice

$\det(X^\top X) \neq 0$ and $\operatorname{rank}(X) = 3$; exactly one $w$ minimises the error, since $50/10 \neq 59/15$ shows the two temperature columns are not multiples of each other.

$\det(X^\top X) = 0$ and $\operatorname{rank}(X) = 2$; no $w$ minimises the error — the least-squares problem has no solution at all.

$\det(X^\top X) = 0$ and $\operatorname{rank}(X) = 1$; exactly one $w$ minimises the error.

$\det(X^\top X) = 0$ and $\operatorname{rank}(X) = 2$; infinitely many $w$ minimise the error.