Answer
See below
Work Step by Step
Let $A$ be an upper triangular matrix
$A=\begin{bmatrix}
a_{11} & ... & a_{1n}\\
0 & ... & .\\
0 & ... & a_{nn}
\end{bmatrix}$
$B$ be a lower triangular matrix
$B=\begin{bmatrix}
b_{11} & ... & 0\\
b_{21} & ... & .\\
b_{n1} & ... & b_{nn}
\end{bmatrix}$
$\Rightarrow A+B=\begin{bmatrix}
a_{11}+b_{11} & ... & a_{1n}\\
b_{21} & ... & .\\
b_{n1} & ... & a_{nn}+b_{nn}
\end{bmatrix}$
And we know that sum of two rational numbers is neither upper nor lower triangular matrix.
$\Rightarrow A+ B$ is not closed under addition.
Set of Rational numbers is closed under usual addition.
Because our scalars can come from set of real numbers
In particular, let $\lambda$ be a scalar.
$\Rightarrow \lambda.A=\lambda\begin{bmatrix}
a_{11} & ... & a_{1n}\\
0 & ... & .\\
0 & ... & a_{nn}
\end{bmatrix}=\begin{bmatrix}
\lambda a_{11} & ... & \lambda a_{1n}\\
0 & ... & .\\
0 & ... & \lambda a_{nn}
\end{bmatrix}$
Therefore set of rational number is closed under addtion.
