Answer
See below
Work Step by Step
Let $A,B \in \mathbf{Q}$
$\Rightarrow A,B$ have zeros below main diagonal
And we know that sum of two rational numbers is componentwise
$\Rightarrow A+ B$ is 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.
Let $a_{ij}=0 \in \mathbf{Q}$
$\Rightarrow \lambda .a_{ij}=\lambda.0=0$
Therefore set of rational number is closedunder addtion.
