Let $B = \left(\begin {array}{ccc|c} a & b & c. & d \\ 0 & e & f & g\\ 0 & 0 & h & i \end{array}\right)$ \ be a $\text{REF}$ that belongs to a non-homogenous linear system.
$(1)$ If $ i \ne 0 $, then, $ \det(B) = \begin{vmatrix} a & b & c \\ 0 & e & f \\ 0 & 0 & h \end{vmatrix}$ $\ne 0$.
$\quad \quad$ False. We have that $\det(B) = aeh $, so $i$ does not affect the value of the determinant. Even if $ i \ne 0,$ $ \det(B) \ne 0 $ when $ a,e,h \ne 0 $
$(2)$ If the $3^{\text{rd}}$ column is a pivot, then the system has a unique solution.
$\quad \quad$ False. Let $ c,d \ne 0$, and $a,b,e,f,g,h,i = 0.$ The system has infinitely many solutions.
I am not sure if my reasons are correct for $(1),$ does $i$ indicate any property of the matrix, which would change the determinant?