Let 


$ E $ - Elliptic curve defined over $ {\mathbb{Q}} $. 


$G\_{\mathbb{Q}}$ - The absolute Galois group, $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}) $ of $\mathbb{Q}$. 


$ E[3] $ - $3$-torsion points of $ E $.


Suppose $ \rho$ denotes the $ G\_{\mathbb{Q}} $-representation associated to $ E[3]$. 
If $ E $ has a $3$-torsion point over $ \mathbb{Q} $, how to prove
 $$
 \rho \sim
 \left( {\begin{array}{cc}
 1 & \eta \\
 0 & \chi \\
 \end{array} } \right)
 $$ where $ \chi $ is the mod 3 cyclotomic character $?$



Galois representation attached to $3$-torsion points of an elliptic curve
Assuming that $E$ has a rational $3$-torsion point, that point must be fixed, and hence your representation takes the form
$$ \rho \sim \left( \begin{matrix} 1 & \eta\_{i} \\ 0 & \chi \end{matrix}\right)$$
as you say. It's a general fact that the determinant of the $G\_{\mathbb{Q}}$ representation attached to an elliptic curve is the cyclotomic character. This follows because the Weil pairing is bilinear, alternating, and Galois-invariant. (See Proposition III.8.1 in Silverman's Arithmetic of Elliptic Curves.)


